text:
\section{Introduction}
The dual purpose of this paper is to study Hilbert's fourth problem in the compact setting as well as to explore the concept of geodesic reversibility introduced by Bryant in~\cite{Bryant:2006}.

We recall that Hilbert's fourth problem, in its succinct formulation due to Busemann~(\cite{Busemann:Hilbert}), asks {\it to construct and study all (possibly) asymmetric continuous metrics on open convex subsets of real projective $n$-space for which oriented projective lines are geodesics}. When the distances are symmetric, and the orientation of the lines is irrelevant, Busemann~(\cite{Busemann:1961b}) gave a simple integral-geometric construction for these {\it projective metrics} and Pogorelov showed in~\cite{Pogorelov:1979} that every symmetric projective metric can be obtained as a limit of those constructed by Busemann. The reader will find an enticing history of the problem in~\cite{Papadopoulos:2014}.

In this paper we show that the problem of constructing all asymmetric projective metrics, which remains open in the case of open convex subsets of ${\mathbb R}^n$, has a surprisingly rigid answer on real projective $n$-space:

\begin{maintheorem}{I} \label{spherical-space-forms}
A smooth projective Finsler metric on the real projective space ${\mathbb {RP}}^n$ is the sum of a reversible
projective Finsler metric and the differential of a smooth function on ${\mathbb {RP}}^n$. Moreover,
 if $d$ is a continuous asymmetric projective metric on ${\mathbb {RP}}^n$, there exists a continuous function $f : {\mathbb {RP}}^n \rightarrow {\mathbb R}$ such that $d(x,y) - d(y,x) = f(y) - f(x)$ for every pair of points $x,y \in {\mathbb {RP}}^n$.
\end{maintheorem}

In other words, the {\it only} way to construct a continuous asymmetric
distance function $d$ on ${\mathbb {RP}}^n$ for which all oriented projective lines are geodesics is to take a  symmetric distance function  $\Tilde{d}$  with the same property, find a continuous function
$f: {\mathbb {RP}}^n \rightarrow {\mathbb R}$ such that
$$
\Tilde{d}(x,y) > |f(y) - f(x)|
$$
for all pairs of distinct points, and set
$$
d(x,y) = \Tilde{d}(x,y) + f(y) - f(x) .
$$

The two-dimensional case of the first part of Theorem~\ref{spherical-space-forms} is due to Gautier Berck who in 2012 privately communicated a proof---based on the range characterization of an integral-geometric transform---that a smooth projective Finsler metric on the $2$-sphere is the sum of a reversible Finsler metric and the differential of a function on $S^2$. Berck's proof is strictly two-dimensional, but armed with the knowledge that such rigidity was possible the author was able to find a different proof that works in all dimensions and applies to a much larger class of Finsler metrics.

\begin{definition}[Bryant~\cite{Bryant:2006}]
A Finsler metric is said to be {\it geodesically reversible} if any geodesic can be reparametrized in an orientation-reversing way so as to remain a geodesic.
\end{definition}

\begin{maintheorem}{II} \label{Zoll}
If $(M,F)$ is a smooth, geodesically-reversible Zoll Finsler manifold, then $F$ is the sum of a reversible Zoll Finsler metric and the differential of a smooth function on $M$.
\end{maintheorem}

A proof of this result was sketched in the preprint~\cite{Alvarez:2013} of which this paper is the final version. However, the underlying cause for this rigidity was still mysterious. Indeed, there
are many examples of geodesically reversible or even projective Finsler metrics on open subsets of ${\mathbb R}^n$  (e.g., asymmetric norms in ${\mathbb R}^n$, the Funk metric in the interior of a convex body) that
are not sums of reversible metrics plus closed $1$-forms, but on compact manifolds it seems that the only such examples are all flat. This gave rise to the project of understanding projective and other geodesically-reversible Finsler metrics on compact manifolds. Since the only manifolds with a projective structure (i.e., a $(PGL(n+1,{\mathbb R}), {\mathbb {RP}}^n)$-structure) which are known to carry a $C^2$ projective Finsler metric are also those that carry a projective Riemannian metric (necessarily of constant curvature), this led to main problem tackled in~\cite{Alvarez-Barbosa:2018} and in the present work:  {\it to describe all $C^2$ projective Finsler metrics defined on compact or finite-volume space forms.}

Note that the Busemann-Pogorelov solution of Hilbert's fourth problem in the reversible setting together with Theorem~\ref{spherical-space-forms} provides a solution for spherical space forms, even when  considering continuous distance functions. For compact Euclidean space forms the problem was solved by Alvarez Paiva and Barbosa Gomes in~\cite{Alvarez-Barbosa:2018} by showing that all such metrics are the sum of a flat Finsler metric and a closed $1$-form. In the hyperbolic case we have the following two results:

\begin{maintheorem}{III} \label{hyperbolic-space-forms-non-reversible}
 In two and three dimensions, every $C^2$ projective Finsler metric on a hyperbolic space form of finite volume is the sum of a multiple of the hyperbolic metric and a closed $1$-form.
\end{maintheorem}

 \begin{maintheorem}{IV} \label{hyperbolic-space-forms-reversible}
If $F$ is a reversible $C^2$ projective Finsler metric on a hyperbolic space form of finite volume, then $F$ is a multiple of the hyperbolic metric.
\end{maintheorem}

Like in the case of Euclidean space forms treated in~\cite{Alvarez-Barbosa:2018}, and unlike
the spherical case, the regularity of the metric is important: examples by Busemann, Saltzmann, and
Bonahon~(\cite{Busemann-Salzmann:1965} and~\cite{Bonahon:1993}) show that there are plenty of non-proportional, continuous, symmetric projective metrics on hyperbolic surfaces. However, like in the case of spherical space forms, the results for hyperbolic space forms of finite volume follow from  results on geodesic-reversibility and projective rigidity for a more general class of Finsler metrics.

\begin{maintheorem}{V}\label{geodesic-reversibility}
Let $M$ be a two- or three-dimensional manifold and let $F$ be a $C^2$ geodesically reversible Finsler metric on $M$. If the geodesic flow does not admit any non-trivial continuous integral of motion, then the metric $F$ is necessarily the sum of a reversible metric and a closed 1-form.
\end{maintheorem}

\begin{maintheorem}{VI} \label{transitive-geodesic}
Let $F_1$ and $F_2$ be two reversible $C^2$ Finsler metrics defined on a connected manifold $M$ and which have the same unparametrized geodesics. If the geodesic flow of either metric does not admit any non-trivial continuous integral of motion, then the metric $F_2$ is necessarily a  constant multiple of $F_1$.
\end{maintheorem}

Some examples of Finsler manifolds whose geodesic flow does not admit any non-trivial continuous integrals of motion are (1) compact Finsler manifolds with negative flag curvature (Foulon~\cite{Foulon:1992}), (2) compact Finsler surfaces of genus greater than one without conjugate points (Barbosa Gomes and Ruggiero~\cite{Barbosa-Ruggiero:2013}), and (3) compact Finsler manifolds without conjugate points and whose universal cover satisfies the uniform Finsler visibility condition (Chimenton, Barbosa Gomes, and Ruggiero~\cite{Barbosa-Chimenton-Ruggiero:2020}).

Theorem~\ref{transitive-geodesic} is the generalization to reversible Finsler metrics of a result by Matveev and Topalov (see Corollary~1 in~\cite{Matveev-Topalov:2000, Matveev-Topalov:2003}). Note however that both the local and global study of Riemannian metrics with the same unparametrized geodesics is full of rigidity phenomena that do not carry over to Finsler metrics.   For instance, a simple consequence of Corollary~2 in~\cite{Matveev-Topalov:2003} is that {\it two Riemannian metrics in a connected manifold that have the same unparametrized geodesics and agree on some open set are necessarily identical.} This is false for Finsler metrics, even reversible ones: the Busemann construction of projective metrics on the sphere allows us to construct an abundance of non-isometric examples of smooth reversible Finsler metrics all of whose geodesics are great circles and which agree with the standard Riemannian metric in the complement of any fixed neighborhood of  the equator (see Section~\ref{Zoll-metrics}). Nevertheless, we do have the following rigidity phenomenon:

\begin{maintheorem}{VII} \label{continuation-principle}
Let $F_1$ and $F_2$ be two $C^2$ Finsler metrics with the same unparametrized geodesics. If the velocity field of every geodesic passes through some open set in the tangent bundle where $F_1$ equals $F_2$, then the metrics agree everywhere modulo the addition of a closed $1$-form.
\end{maintheorem}

An interesting and surprising aspect of the proofs of Theorems~\ref{Zoll}, ~\ref{geodesic-reversibility}, and~\ref{transitive-geodesic}  is that they are
based on the study of volumes and areas on Finsler spaces (see~\cite{Thompson:1996} for a thorough account of the convex-geometric origins of the subject and~\cite{Alvarez-Thompson:2004} for a more Finsler-oriented and updated account). In the final analysis, Theorem~\ref{Zoll} rests on the Brunn-Minkowski theorem, Theorem~\ref{geodesic-reversibility} on Chakerian's characterization of bodies of constant width and constant brightness in relative geometry~(\cite{Chakerian:1967}), and Theorem~\ref{transitive-geodesic} on the solution of the Minkowski problem. Despite---or perhaps due to---the interplay between symplectic, integral and convex geometry on which they are based, the proofs are exceedingly simple. In Sections~\ref{volumes} and~\ref{Hilbert-form} we review some basic concepts and develop an adequate language in which to describe this interplay.
Section~\ref{Hilbert-form} also contains the proof of Theorem~\ref{continuation-principle}. Sections~\ref{Zoll-metrics} and~\ref{geodesic-rigidity} contain the proofs of Theorems~\ref{spherical-space-forms} to~\ref{transitive-geodesic}.

\medskip
\noindent {\bf Acknowledgments.} The author thanks G.~Berck, A.~Thompson, J.~Barbosa~Gomes, S.~Tabachnikov, B.~McKay, and V.~Matveev for the many interesting discussions that have in some form
or another found their way to the following pages. I'm specially grateful to R.~Howard and D.~Ryabogin for their timely answers to my naive questions about bodies of constant width and constant brightness, and to R.~Ruggiero for his comments on the first version of this work.

\section{Volumes and areas in Finsler spaces} \label{volumes}

A {\sl continuous Finsler metric} on a manifold $M$ is a continuous function $F:TM \rightarrow [0,\infty)$ such that its restriction to each tangent space $T_x M$ $(x \in M)$ is a (possibly) asymmetric norm. In order to study things like geodesic flows, curvature, and differential invariants, it is usual to ask for more regularity and a stricter notion of convexity:

\begin{definition}
Let $V$ be a finite-dimensional vector space over the reals. An asymmetric norm $\|\cdot\| : V \rightarrow [0,\infty)$ is said to be a {\sl Minkowski norm} if  outside the origin the function $\|\cdot\|^2$ is $C^2$ and its Hessian is positive-definite. A continuous function
$F : T M \rightarrow [0, \infty)$ defined on the tangent bundle of a smooth manifold $M$ will be said to be a $C^k$ $(k \geq 2)$ {\sl Finsler metric} if it is $C^k$ outside the zero section and its restriction to each tangent space is a Minkoswki norm.
\end{definition}

Note that the restriction of the Finsler metric $F$ to a tangent space $T_xM$ is the support function of the {\sl unit co-disc}
$$
D_x^* M := \{\xi_x \in T_x^* M: \xi_x(v_x) \leq 1 \text{ whenever } F(v_x) \leq 1 \}.
$$
The {\sl unit co-disc bundle} $D^* M \subset T^*M$ of a Finsler metric on a manifold $M$ is the union of all the co-discs $D_x^* M$, $x \in M$, seen as a disc bundle over $M$.

\begin{definition}
The {\sl Holmes-Thompson volume} of a Finsler metric on a manifold $M$ of dimension $n$ is the symplectic volume of its unit co-disc bundle divided by the volume of the $n$-dimensional Euclidean unit disc. The {\sl area} of a submanifold is defined as the Holmes-Thompson volume of the submanifold provided with its inherited metric.
\end{definition}

It is also possible to define the Holmes-Thompson volume by considering its volume density---the function $\Phi : \bigwedge^n T M  \to [0,\infty)$ that measures the volume of parallelotopes formed by vectors tangent to $M$---:  if $v_1, \ldots, v_n $  is a basis of $T_xM$, the $n$-vector
$v_1 \wedge \cdots \wedge v_n$ is a volume form on $T_x^* M$. Denoting the volume of the $n$-dimensional Euclidean unit disc by $\varepsilon_n$, we define
$$
\Phi(v_1 \wedge \cdots \wedge v_n) :=  \frac{1}{\varepsilon_n} \int_{D_x^* M} |v_1 \wedge \cdots \wedge v_n| .
$$
The equivalence between both definitions follows immediately from the computation of the
pushfoward of the symplectic volume form on $D^*M$ onto $M$ under the canonical projection
$\pi : D^* M \rightarrow M$.

From this second description it is clear that adding a $1$-form to a Finsler metric does not alter its volume density: adding the form merely translates the unit co-discs in each cotangent space. A more interesting property of the Holmes-Thompson volume is its behavior with respect to {\sl central symmetrization}:

\begin{definition}
The central symmetrization of a Finsler metric  $F$ is the Finsler metric
defined by $F^S(v_x) = (F(v_x) + F(-v_x))/2$.
\end{definition}

\begin{proposition}\label{central-symmetrization}
The Holmes-Thompson volume density of a Finsler metric on a manifold $M$ is less than or equal to that of its central symmetrization. Moreover, the volume densities are equal if and only if $F = F^S + \beta$ for some $1$-form $\beta$ on $M$.
\end{proposition}

\proof
Note that if $D_x^* M \subset T^*M$ is the convex body supported by the restriction of $F$ to $T_xM$, the restriction of $F^S$ to this vector space is the support function of the symmetrized body
$\Delta D_x^*M := (D_x^* M - D_x^* M)/2$. A well-known consequence of the Brunn-Minkowski inequality is that the volume of a convex body $K$ is less than or equal to that of its symmetrization $\Delta K = (K - K)/2$ with equality if and only if $K$ is symmetric with respect to a (unique) point $p$ in its  interior. Applying the theorem at every cotangent space $T^*_x M$, $x \in M$, concludes the proof. Note that the $1$-form will be smooth if the bodies $D_x^*M$ depend smoothly on the base point.
\qed

\subsection{Areas in Finsler spaces}

The extrinsic description of areas in Finsler spaces requires us to define the area of each $k$-dimensional parallelotope in $T_x M$ at every point $x \in M$. Given a decomposable $k$-vector $a$ formed by taking the exterior product of linearly independent tangent vectors $v_1,\ldots,v_k$
in $T_xM$, consider the subspace $\langle a \rangle$ spanned by these vectors together the natural inclusion $i_a : \langle a \rangle \rightarrow T_xM$.  The dual map  $i_a^* : T_x^* M \rightarrow \langle a \rangle^*$ projects the unit co-disc $D_x^*M \subset T_x^*M$
to a convex body $i_a^*(D_x^*M)$ in $\langle a \rangle^*$. Since $a$ is a volume form in  $\langle a \rangle^*$, we are able to define the {\sl Holmes-Thompson $k$-area density}
$$
\varphi_k(a_x) = \frac{1}{\varepsilon_k} \int_{i_a^*(D_x^*M)} |a| .
$$

When the Finsler manifold is ${\mathbb R}^n$ provided with a norm with unit ball $D$ the Holmes-Thompson $k$-area density can be identified with the {\it $k$-th projection function} of the polar body $D^\circ$~(\cite[Chapter VI]{Thompson:1996}). Since this provides some intuition and allows us to connect the theory of areas in Finsler spaces with important results in convex geometry, we will go carefully over the identification.

Consider ${\mathbb R}^n$ provided with its standard inner product together with a convex body $D$ containing
the origin in its interior. The dual space ${\mathbb R}^{n*}$ is identified with ${\mathbb R}^n$ via the inner product, and the dual body $D^*$ is identified with the polar body
$$
D^\circ := \{u \in {\mathbb R}^n : u \cdot v \leq 1  \text{ whenever } v \in D \} .
$$
Given orthonormal vectors $e_1,\ldots,e_k$ in ${\mathbb R}^n$, we set $a := e_1 \wedge \cdots \wedge e_k$
and denote the subspace they span by $\langle a \rangle$. Using the restriction of the inner product
on ${\mathbb R}^n$ to $\langle a \rangle$ we can identify it with its dual space $\langle a \rangle^*$. With these identifications, the map  $i_a^* : {\mathbb R}^{n*} \rightarrow \langle a \rangle^*$ used
in the definition of the $k$-area density becomes the orthogonal projection from ${\mathbb R}^n$ to  $\langle a \rangle$. {\it In conclusion, taking all these identifications into account, $\varphi_k(a)$ is simply the Euclidean area of the orthogonal projection of $D^\circ$ onto  $\langle a \rangle$ divided by the area of the $k$-dimensional unit disc in Euclidean space.}

This geometric intuition makes it obvious that adding a $1$-form to $F$ or considering the {\it reverse metric} $F_{-}$, defined by $F_{-}(v_x) = F(-v_x)$, does not change the $k$-area densities. The following results also follow from the relation between the Holmes-Thompson $k$-area densities and $k$-th projection functions in convex geometry. An excellent reference for what follows is Chapter~3 of~\cite{Gardner:1995}.

\begin{proposition}\label{areas-determine-metric}
Given a Finsler metric $F$ on a manifold $M$, there exists a unique reversible Finsler metric
$F^B$ on $M$, the areal symmetrization of $F$, that defines the same $(n-1)$-area density.
\end{proposition}

\proof
A classic result in convex geometry states that given a convex body $K \subset {\mathbb R}^n$, there is a unique $0$-symmetric body $\nabla K$ called the {\it Blaschke body} of $K$ such that the areas of orthogonal projections of $K$ and $\nabla K$ onto any fixed hyperplane are the same. Applying this
construction to every unit co-disc we obtain a  reversible Finsler metric $F^B$ whose restriction to every tangent space $T_xM$ is the support function of the Blaschke body of $D_x^*M$. By construction $F$ and $F^B$ have the same $(n-1)$-area density.
\qed

\begin{proposition}
The Holmes-Thompson volume density of a Finsler metric $F$ on a manifold $M$ is less than
or equal to that of its areal symmetrization $F^B$. Moreover, the volume densities are equal if
and only if $F = F^B + \beta$ for some $1$-form $\beta$ on $M$.
\end{proposition}

\proof
By the Kneser-S\"uss inequality (Theorem~3.3.9 in~\cite[p. 109]{Gardner:1995}), if $K \subset {\mathbb R}^n$ is a convex body, its volume is less than or equal to the volume of its Blaschke body $\nabla K$ and equality holds if and only if $K$ is centrally symmetric. The result follows by applying this inequality to every unit co-disc of the Finsler manifold $(M,F)$ in the same way we applied the Brunn-Minkowski inequality in Proposition~\ref{central-symmetrization}.
\qed

\begin{proposition}\label{constant-width-constant-brightness}
Given a $C^2$ Finsler metric $F$ on a three-dimensional manifold $M$, its central symmetrization  $F^S$ is conformal to its areal symmetrization $F^B$ (i.e., $F^S = e^f F^B$, where $f$ is a continuous function on $M$) if and only if $F = F^S + \beta$ for some $1$-form $\beta$ on $M$.
\end{proposition}

\proof
Let us fix an arbitrary point $x \in M$ and consider what the hypotheses tell us about the convex body $D_x^* M$. Since $F_S$ and $F_B$ are conformal, their restrictions to $T_xM$ are multiples and
thus the central symmetrization and the Blaschke body of $D_x^* M$ are dilates:
$\Delta D_x^*M = \lambda \nabla D_x^*M$ for some $\lambda > 0$. The key point in the proof is to remark that  $D_x^* M$ is a body of constant width with respect to $\Delta D_x^*M$ and a body of constant brightness with respect to $ \nabla D_x^*M$, which implies that $D_x^*M$ is a body of constant width and constant brightness relative to the body $\Delta D_x^* M$. Since by hypothesis the boundary of  $\Delta D_x^* M$ is $C^2$ and quadratically convex, the main result of Chakerian~\cite{Chakerian:1967} (see also~\cite{Howard:2006}) implies that $D_x^*M$ is centrally symmetric. This is true for every point $x \in M$ and hence $F$ is the sum of a reversible metric and a $1$-form.
\qed

The generalization of Chakerian's result in~\cite{Chakerian:1967} to dimensions greater than three is
a long-standing open problem in convex geometry. As we shall see in Section~\ref{geodesic-rigidity},
should the generalization prove true, Theorem~\ref{geodesic-reversibility} would also generalize to arbitrary dimension.

\subsection{Basic integral-geometric formulas}

Let $(M,F)$ be a $C^2$ Finsler manifold and let $S^* M := \partial D^*M$ be its {\it unit co-sphere bundle}. The pullback of the canonical $1$-form $\alpha$ in $T^*M$ to the co-sphere bundle is a $C^1$ contact form and its Reeb vector field defines the geodesic (local) flow of $(M,F)$.

Assuming the metric is $C^3$ and the space of oriented geodesics of $(M,F)$ is itself a manifold $\geod{M}$,  it is well-known (cf.~\cite[p.~48--49]{Arnold-Givental:1990}) that $\geod{M}$  carries a symplectic $2$-form $\omega_F$ defined as follows: if
$$
\begin{CD}
S^*M   @>i>>  T^*M \\
@V{\pi}VV          \\
\geod{M}
\end{CD}
$$
are the canonical projection onto $\geod{M}$ and the canonical inclusion
into $T^{*}M$, $\omega_F$ is the unique $2$-form satisfying the equation $\pi^{*}\omega_F = i^{*}d\alpha$. This is standard symplectic reduction and, as explained in Theorem~5.3.23 in~\cite[p.~416]{Abraham-Marsden}, it requires the symplectic form $d\alpha$ to
be $C^1$, which in turn requires the metric $F$ to be $C^3$. However, with very little effort can
get by with less. Employing the notation above, we have the following

\begin{theorem}\label{space-of-geodesics}
Let $(M,F)$ be a $C^2$ Finsler manifold and assume the space of geodesics of $(M,F)$ is itself
a manifold $\geod{M}$. There is a unique continuous, maximally non-degenerate $2$-form $\omega_F$
on $\geod{M}$ that satisfies the equation $\pi^{*}\omega_F = i^{*}d\alpha$.
\end{theorem}

The key point is to show that the continuous form $i^{*}d\alpha$ is invariant under the geodesic
flow without being able to take Lie derivatives. Since a slight generalization of this is also the key to Theorem~\ref{continuation-principle}, we encapsulate what we need in one result.

\begin{lemma}\label{surfaces-of-section}
Let $P$ be a contact manifold with $C^1$ contact form $\alpha$ and let  $S_1, S_2 \subset P$ be two hypersurfaces that are transverse to its Reeb vector field. If every Reeb orbit that intersects $S_1$ also intersects $S_2$ at a later time, then the map  $\tau: S_1 \rightarrow S_2$  that sends each point $p \in S_1$ to the first crossing of $S_2$ and the
Reeb orbit with initial condition $p$ is an isomorphism between $(S_1, d\alpha)$ and $(S_2,d\alpha)$. \end{lemma}

\proof
It is enough to prove that the integral of $d\alpha$ over every sufficiently small $2$-disc $R \subset S_1$ equals the integral of $d\alpha$ over its image $\tau(R)$. In order to do this, consider
the cylinder $C \subset P$ formed by the pieces of Reeb orbits joining the points in the boundary of $R$ with those in the boundary of $\tau(R)$. Taking into account that tangent vectors to  Reeb orbits are in the kernel of $d\alpha$ and applying Stokes formula (twice) we have that
$$
0 = \int_C d\alpha = \int_{\partial R} \alpha - \int_{\partial \tau(R)} \alpha =
\int_R d\alpha - \int_{\tau(R)} d\alpha .
$$
\qed

\proof[Proof of Theorem~\ref{space-of-geodesics}]
Given a sufficiently small open set $U \subset \geod{M}$ take a hypersurface $\Sigma \subset S^*M$ that is transverse to the flow and projects diffeomorphically onto $U$. Define $\omega_F$ by
setting $\pi^* \omega_F$ to be equal to $d\alpha$ on $\Sigma$. By Lemma~\ref{surfaces-of-section}, the choice of transverse hypersurface is immaterial. The form $\omega_F$ is maximally non-degenerate because  $\pi^* \omega_F^{n-1}$ is equal to the nowhere-zero form $d\alpha^{n-1}$ on $\Sigma$.
\qed

Among the many examples of Finsler manifolds whose space of geodesics is
a manifold we find convex neighborhoods in Finsler spaces,
Hadamard Riemannian manifolds (for both of these examples see
\cite{Ferrand:1997}), Zoll metrics (\cite{Besse:1978}), Minkowski geometries, Hilbert geometries,
Funk geometries and, more generally, projective Finsler metrics (\cite{Alvarez:2005}
and \cite{Alvarez-Fernandes:1998}). For these manifolds we have the following
integral-geometric formulas:

\begin{theorem}[\cite{Alvarez-Berck:2006}]\label{crofton-hypersurface}
Let $M$ be an $n$-dimensional $C^2$ Finsler manifold with manifold of
geodesics $\geod{M}$. If $\omega_F$ is the induced $2$-form on $\geod{M}$
and  $N \subset M$ is any immersed hypersurface, then
$$
{\rm vol}_{n-1}(N) = \frac{1}{2\epsilon_{n-1}(n-1)!} \, \int_{\gamma \in \geod{M}}
\#(\gamma \cap N) |\omega_F^{n-1}| ,
$$
where $\epsilon_{n-1}$ is the volume of the Euclidean unit ball of dimension
$n-1$.
\end{theorem}

\begin{proposition}[Santal\'o formula]
Let $F$ be a $C^2$ Finsler metric on an $n$-dimensional manifold $M$. If $f : S^*M \rightarrow {\mathbb R}$ is a continuous function on its unit co-sphere bundle, then
$$
\int_{S^*M} f \alpha \wedge d\alpha^{n-1} =
\int_{l \in \geod{M}} \! \! \left( \int_l f \alpha \right) \omega_F^{n-1} .
$$
In particular, taking $f \equiv 1$,
$$
{\rm vol}_n(M,F) = \frac{1}{\varepsilon_n n!} \int_{S^*M} \! \! \alpha \wedge d\alpha^{n-1}
= \frac{1}{\varepsilon_n n!} \int_{l \in \geod{M}} \! \! {\rm length}(l) \, \omega_F^{n-1} .
$$
\end{proposition}

\proof
Using that the tangents to the fibers of the projection $\pi : S^*M \rightarrow \geod{M}$ are in
the kernel of the form $d\alpha^{n-1} = \pi^* \omega_F^{n-1}$ we obtain the identity
$$
\pi_* \, f \alpha \wedge d\alpha^{n-1} = \left( \int_l f \alpha \right) \omega_F^{n-1} ,
$$
which immediately implies the result.
\qed

\section{The Hilbert form and geodesic reversibility} \label{Hilbert-form}

A common feature of differential $1$-forms and Finsler metrics is that the result of integrating them over a continuously differentiable curve is independent of the parameterization as long as the orientation of the curve is unchanged. In the case of Riemannian and reversible Finsler metrics, the integral is also invariant under a change of orientation.  General integrands that satisfy these properties are known, respectively, as $1$-densities and even $1$-densities:

\begin{definition}
A continuous function $L : T M \rightarrow {\mathbb R}$ defined on the tangent bundle of a smooth manifold $M$ will be said to be a $1$-{\sl density} of class $C^k$ $(k \geq 1)$
if it is  $C^k$ outside the zero section and positively homogeneous of degree $1$ in the velocities (i.e., $L(tv_x) = tL(v_x)$ for $t > 0$). A $1$-density $L$
will be called  {\sl even} if it  satisfies $L(-v_x) = L(v_x)$ and  {\sl odd} if it satisfies $L(-v_x) = -L(v_x)$.
\end{definition}

Note that $C^k$ Finsler metrics form a convex cone inside the vector space of $C^k$ densities.
Moreover, every Finsler metric $F$ can be uniquely decomposed into odd and even densities:  if $F_{-}$ denotes the reverse of the Finsler metric $F$, we can write
$$
F = \frac{1}{2}(F - F_{-})  + \frac{1}{2}(F + F_{-}) =:  F^{\mathcal{O}} + F^S  .
$$

\subsection{The Hilbert Form}
In what follows it will be convenient to adopt a manifestly covariant formalism for the calculus of variations due to Hilbert (see the text of his 23rd problem in~\cite{Hilbert:problems}). In modern terminology it is quite simple (cf.~\cite{Chern:1996}): given a $C^2$ $1$-density $L$ on $M$ we  define a bundle map $\delta L : {TM \! \setminus \! \mathbf{0}} \rightarrow T^* M$ from the tangent bundle of $M$ minus its zero section to the cotangent bundle of $M$
by the formula
$$
\delta L(v_x) \cdot w_x := \frac{d}{dt}L(v_x + tw_x)\restrict{t = 0} \, .
$$
The map $\delta L$ is used to pull back the canonical $1$-form $\alpha$ on the cotangent bundle to a
$1$-form $\alpha_{{}_L} := (\delta L)^* \alpha$ on ${TM \! \setminus \! \mathbf{0}} $.  This is {\it the Hilbert form} associated to $L$.  In local coordinates
$$
\alpha_{{}_L} = \frac{\partial L}{\partial v_1} \, dx_1 +
\cdots + \frac{\partial L}{\partial v_n} \, dx_n .
$$

Two immediate consequences of this local expression are that (1) the Hilbert $1$-form $\alpha_L$ is homogeneous of degree zero and so descends naturally onto the space of tangent directions $STM := ({TM \! \setminus \! \mathbf{0}}) / {\mathbb R}^+$, and (2) that  the Hilbert form of a differential $1$-form $\beta \in \Omega^1(M)$, seen as the odd $1$-density $(x,v) \mapsto \beta_x(v)$, is just the pullback of  $\beta$ to ${TM \! \setminus \! \mathbf{0}}$ under the canonical projection $\pi : {TM \! \setminus \! \mathbf{0}}  \rightarrow M$.

Using Euler's formula for homogeneous functions we see at once that $L(x,v) = \alpha_{(x,v)}(v)$. In order to be precise, we need to interpret the symbol $v$ in two different ways: as a vector in $T_x M$ and as a vector in $T_{(x,v)}({TM \! \setminus \! \mathbf{0}})$ (i.e., as the velocity  vector at $t = 0$ of the curve
$(x,v + tv)$ in the tangent bundle of $M$). It follows that for any continuously differentiable curve $\gamma$ with velocity curve $\gamma'$
$$
 \int_\gamma L = \int_{{\gamma}'} \alpha_{{}_L} .
$$

Using the Hilbert form, the Euler-Lagrange equations  for $1$-densities can be elegantly formulated as follows:

\begin{theorem}\label{first-variation}
A smooth curve $\gamma : {\mathbb R} \rightarrow M$ with nowhere vanishing velocity is extremal for the variational problem $\gamma \mapsto \int_\gamma L$
if and only if its second derivative $\gamma''$, seen as  the velocity of the curve $\gamma'$, satisfies the equation
$$
d \alpha_{{}_L} (\gamma''(t), \cdot) = 0
$$
for all values of $t$.
\end{theorem}

\proof
Given  a smooth, compactly-supported variation $\gamma_s$ of the curve $\gamma = \gamma_0$ we wish to compute the derivative at $s = 0$ of the
function
$$
s \longmapsto \int_{\gamma_s} L = \int_{\gamma'_s} \alpha_{{}_L}  .
$$
Writing $\gamma'_s = \phi_s(\gamma')$, where $\phi_s : {TM \! \setminus \! \mathbf{0}} \rightarrow {TM \! \setminus \! \mathbf{0}}$ is a smooth isotopy that equals the identity outside a compact set, we
have that
$$
\frac{d}{ds}\left( \int_{\gamma'_s} \alpha_{{}_L} \right) \restrict{s = 0} = \frac{d}{ds} \left( \int_{\phi_s(\gamma')} \alpha_{{}_L} \right) \restrict{s = 0} =
\int_{\gamma'} \mathcal{L}_X \alpha_{{}_L} .
$$
Here $\mathcal{L}_X$ is the Lie derivative with respect to the  vector field on ${TM \! \setminus \! \mathbf{0}}$ defined by $X(v_x) = d\phi_s(v_x)/ds\restrict{s = 0}$. Using Cartan's formula for the Lie derivative and assuming that either $\gamma$ is periodic or that the vector field $X$ is supported away from the endpoints of $\gamma'$,
$$
\int_{\gamma'} \mathcal{L}_X \alpha_{{}_L} = \int_{\gamma'} d\alpha_{{}_L}(X, \cdot) + d(\alpha_{{}_L}(X)) =  \int_{\gamma'} d\alpha_{{}_L}(X, \cdot) \, .
$$
If $\gamma$ is an extremal, this quantity must be zero for an arbitrary compactly-supported vector field $X$ and, therefore, we have that
$d \alpha_{{}_L} (\gamma''(t), \cdot) = 0$ for all values of $t$.
\qed

A direct consequence of this theorem---and arguably the most important remark in the study
of inverse problems for $1$-densities and Finsler metrics---is the following {\it principle of superposition:}

\begin{corollary}
Given a family of smooth oriented curves on a manifold $M$, the set of $1$-densities on $M$ for which all the curves in the family are extremal is closed under linear combinations. In particular, the set of Finsler metrics on $M$ for which all the curves in the family are geodesic is closed under positive linear combinations.
\end{corollary}

\proof
It is enough to notice that the Hilbert form depends linearly on the $1$-density:
if $L_1$ and $L_2$ are two $1$-densities, then for any real numbers $\lambda$ and $\mu$,
$\alpha_{{}_{\lambda L_1 + \mu L_2}} = \lambda \alpha_{{}_{L_1 }} + \mu \alpha_{{}_{L_2}}$. If a smooth curve $\gamma : {\mathbb R} \to M$ is an extremal
for both $L_1$ and $L_2$, then
$$
d\alpha_{{}_{\lambda L_1 + \mu L_2}} (\gamma'', \cdot) =  \lambda d\alpha_{{}_{L_1 }} (\gamma'', \cdot)  + \mu  d\alpha_{{}_{L_2}}  (\gamma'', \cdot)  = 0 ,
$$
which means it is also an extremal for $\lambda L_1 + \mu L_2$.
\qed

\begin{corollary}
If $F$ is a geodesically reversible $C^2$ Finsler metric, then $F$, its reverse metric $F_{-}$, and its central symmetrization $F^S$ all have the same unparametrized geodesics. Moreover, the geodesics
of $F$ are also extremals of its odd part $F^\mathcal{O}$.
\end{corollary}

Theorem~\ref{first-variation} also implies that every smooth curve with never-vanishing velocity is an extremal for the $1$-density $L$ if and only if $d \alpha_{{}_L}$
vanishes identically. This happens if and only if $L$ is a closed differential form (cf.~\cite[p. 51--59]{Giaquinta-Hildebrandt:vol1}):

\begin{proposition}\label{null-lagrangian}
The exterior differential of the Hilbert form of a  $1$-density $L : T M \rightarrow {\mathbb R}$ vanishes identically if and only if $L$  is a closed $1$-form on $M$.
\end{proposition}

\proof
This is simplest to see in local coordinates:
$$
d \alpha_{{}_L} = d \left( \sum_{i=1}^n \frac{\partial L}{ \partial v_i} \, dx_i \right) = \sum_{1 \leq i,j \leq n} \frac{\partial^2 L}{ \partial v_j \partial v_i} \, dv_j \wedge dx_i
+ \sum_{1 \leq i,j \leq n} \frac{\partial^2 L}{\partial x_j \partial v_i} \, dx_j \wedge dx_i .
$$
If $d \alpha_{{}_L}$ vanishes identically, then both
$$
 \sum_{1 \leq i,j \leq n} \frac{\partial^2 L}{\partial v_j \partial v_i} \, dv_j \wedge dx_i    \  \  \ \hbox{\rm and }   \
  \sum_{1 \leq i,j \leq n} \frac{\partial^2 L}{\partial x_j \partial v_i} \, dx_j \wedge dx_i
$$
vanish identically.  The vanishing of the first expression implies that $L$ is a $1$-form (i.e., that $L$ is linear in the velocities), the  vanishing of
the second  says that this form is closed.
\qed

This in turn leads to a simple proof of the following key result (cf. Crampin~\cite{Crampin:2005}).

\begin{corollary}\label{adding-form}
On a manifold $M$ consider a Finsler metric $F$ and a $1$-form $\beta$. The extremals of $F + \beta$ coincide with those of $F$ if and only if $\beta$ is closed. Moreover, if $F$ is reversible, then $F + \beta$ is geodesically reversible if and only if $\beta$ is closed.
\end{corollary}

\proof
Notice that $d\alpha_{{}_{F + \beta}} = d\alpha_{{}_{F}} + \pi^* d\beta$, where $\pi : {TM \! \setminus \! \mathbf{0}}  \rightarrow M$ is the canonical projection.  It is clear
from Theorem~2.3 that if $\beta$ is closed, $F$ and $F + \beta$ have the same extremals. In order to prove the converse, denote the geodesic
vector field of $(M,F)$ by $X$ and remark that, by hypothesis, it is in the kernel of  both $d\alpha_{{}_{F}}$ and $d\alpha_{{}_{F + \beta}}$. It follows
that $(\pi^* d\beta) (X(v_x),\cdot) = d\beta(v_x, \cdot)$ is identically zero for every vector $v_x \in {TM \! \setminus \! \mathbf{0}}$ and, therefore, $d\beta$ is identically zero.

To establish the second part of the corollary recall that a Finsler metric $L$ is geodesically reversible if and only if $L$ and its reverse have the same extremals. Since $F$ is reversible and $\beta$ is a $1$-form, it follows that $F + \beta$ is geodesically reversible if and only if $F + \beta$ and $F - \beta$ have the same extremals. This implies that the Finsler metric $F$ and the $1$-density $F + \beta$ have the same extremals and $\beta$ must be a closed $1$-form.
\qed

\subsection{Basic geodesic rigidity results}
The two rigidity results that follow are basic in the sense that they depend only on very simple and general assumptions on maximally-degenerate, closed $2$-forms on odd dimensional manifolds. Unlike the other results in this paper, the fact that these closed $2$-forms are the exterior differentials
of Hilbert $1$-forms of Finsler metrics is not fundamental.

\begin{maintheorem}{VII}
Let $F_1$ and $F_2$ be two $C^2$ Finsler metrics with the same unparametrized geodesics. If the velocity field of every geodesic passes through some open set in the tangent bundle where $F_1$ equals $F_2$, then the metrics agree everywhere modulo the addition of a closed $1$-form.
\end{maintheorem}

\proof
Consider the manifold of tangent directions $STM$ together with the Hilbert forms $\alpha_{F_1}$
and $\alpha_{F_2}$, which we consider as $C^1$ contact forms on this space. In light of Proposition~\ref{null-lagrangian}, if we prove that $d\alpha_{F_1}$ and $d\alpha_{F_2}$ are identical, we will have proved that $F_1$ and $F_2$ differ by a closed $1$-form.

Note that since the metrics $F_1$ and $F_2$ have the same unparametrized geodesics, the kernels
of the forms  $d\alpha_{F_1}$ and $d\alpha_{F_2}$ are the same. It suffices then to show that
for any sufficiently small hypersurface $\Sigma \subset STM$ transverse to this common kernel line field, the restrictions of $d\alpha_{F_1}$ and $d\alpha_{F_2}$ coincide.

By hypothesis, if $v_x$ is any given point in $STM$, the Reeb orbits for $\alpha_{F_1}$ and $\alpha_{F_2}$ with initial condition $v_x$ both pass through an open set $U \subset STM$ where $d\alpha_{F_1}$ and $d\alpha_{F_2}$ are  equal. Let $\Sigma \subset STM$ be a transverse hypersurface passing through $v_x$ such that its image $\Sigma_1$ under the time-$t_1$ map of the Reeb flow of $\alpha_{F_1}$ lies entirely in $U$. The surfaces $\Sigma$ and $\Sigma_1$ fall under the hypothesis of Lemma~\ref{surfaces-of-section} for both $(STM,\alpha_{F_1})$ and $(STM,\alpha_{F_2})$. Moreover, since  the map $\tau: \Sigma \rightarrow \Sigma_1$ described in Lemma~\ref{surfaces-of-section}
depends only on the foliation defined by the Reeb orbits and not on their parametrization, it is
the same in both cases. It follows that $\tau$ is both an isomorphism between $(\Sigma, d\alpha_{F_1})$  and $(\Sigma_1, d\alpha_{F_1})$, and between $(\Sigma, d\alpha_{F_2})$ and $(\Sigma_1, d\alpha_{F_2})$. Since $d\alpha_{F_1}$ and $d\alpha_{F_2}$ are equal on  $\Sigma_1 \subset U$, we have that $d\alpha_{F_1}$ and $d\alpha_{F_2}$ are identical on $\Sigma$.
\qed

\medskip
We end the section with a modern formulation, due to Tabachnikov~\cite{Tabachnikov:1999a}, of a key
result in geodesic rigidity which goes back at least to Darboux~\cite[pp.~53--54, Livre~VI, Chapitre~III ]{Darboux:1894}. Since the standard proof for this result involves taking a Lie derivative of the differential of the Hilbert $1$-form---and thus requires that the metric be
$C^3$---, we modify it slightly using Lemma~\ref{surfaces-of-section} so that it works for $C^2$ metrics.

\begin{proposition}\label{integral-of-motion}
If $F_1$ and $F_2$ are two $C^2$ Finsler metrics on an $n$-dimen\-sional manifold $M$ and they have the same unparameterized geodesics, then there exists a  continuous, nowhere-zero function $\nu : {TM \! \setminus \! \mathbf{0}}  \rightarrow {\mathbb R}$ that is homogeneous of degree zero and such that  $d\alpha_{{}_{F_2}}^{n-1} = \nu \, d\alpha_{{}_{F_1}}^{n-1}$. Moreover, $\nu$ is constant along the (common) geodesics of  $(M,F_1)$ and $(M,F_2)$.
\end{proposition}

\proof
Since the forms $d\alpha_{{}_{F_1}}^{n-1}$ and $ d\alpha_{{}_{F_2}}^{n-1}$ are nowhere-zero $(2n-2)$-forms with the same kernel on the $(2n-1)$-dimensional space of tangent directions $STM = ({TM \! \setminus \! \mathbf{0}} )/{\mathbb R}^+$, there exists a continuous, nowhere-zero function $\nu$ on $STM$ such that $d\alpha_{{}_{F_2}}^{n-1} = \nu d\alpha_{{}_{F_1}}^{n-1}$. Extending $\nu$ to ${TM \! \setminus \! \mathbf{0}}$ as a homogeneous function of degree zero proves the first part of the proposition.

In order to show that $\nu$ is constant along orbits of the geodesic flow, take any  hypersurface $\Sigma \subset STM$ transverse to the Reeb flow of $\alpha_{{}_{F_1}}$ and let $\Sigma_t$ be
its image under the time-$t$ map of that flow. Since the Reeb foliations for $\alpha_{{}_{F_1}}$
and $\alpha_{{}_{F_2}}$ are the same, the map $\tau : \Sigma \rightarrow \Sigma_t$ defined in
Lemma~\ref{surfaces-of-section} is the same for both contact forms. It follows that
$$
\tau^* d\alpha_{{}_{F_1}}^{n-1} = d\alpha_{{}_{F_1}}^{n-1} \textrm{ and that }
\tau^* d\alpha_{{}_{F_2}}^{n-1} = d\alpha_{{}_{F_2}}^{n-1} .
$$
This implies that $\nu \circ \tau = \nu$. Since the transverse hypersurface $\Sigma$ and the time
$t$ are arbitrary, this shows that $\nu$ is constant on Reeb orbits.
\qed

\section{Geodesically reversible Zoll metrics} \label{Zoll-metrics}

\begin{definition}
A Finsler manifold is {\sl Zoll} if its geodesic flow is periodic and its prime geodesics all have
the same length. Equivalently, a Finsler manifold is Zoll if its unit co-sphere bundle is a {\it regular contact manifold} in the sense of Boothby and Wang~\cite{Boothby-Wang:1958}.
\end{definition}

In Riemannian geometry, Zoll manifolds are very rare. By a theorem of Berger and Kazdan (see~\cite[pp.~236--246, Appendices~D and~E]{Besse:1978}) the only Zoll Riemannian metric in
${\mathbb {RP}}^n$ is the standard constant curvature metric. It is not known whether there are Zoll Riemannian
metrics in ${\mathbb {CP}}^n$ or ${\mathbb {HP}}^n$ for $n > 1$ besides the standard metrics, and while there are many examples of Zoll metrics on spheres (see~\cite{Besse:1978}), they are not very explicit nor simple to describe. On the other hand there is an abundance of Zoll Finsler metrics. Non-reversible examples are in fact very cheap: take the standard metric in $S^n$, ${\mathbb {RP}}^n$, or
${\mathbb {CP}}^n$ and compose the Hamiltonian of the metric with a homogeneous symplectic transformation that is sufficiently $C^2$ close to the identity. The result will be a (Zoll) Finsler metric whose geodesic flow is symplectomorphic to that of the standard metric.  Reversible examples of Zoll metric take a bit more work, but as we shall see they are still plentiful.

\medskip
\noindent
{\it Reversible projective Finsler metrics on $S^n$ and ${\mathbb {RP}}^n$.} In his work on Hilbert's fourth
problem, Busemann had the lovely idea of inverting the classical Crofton formula for the length of curves on ${\mathbb R}^n$ and projective space: consider a smooth, positive measure $\mu$ on the space of projective hyperplanes in ${\mathbb {RP}}^n$, which is the dual projective space ${\mathbb {RP}}^{n*}$, and define the length of a $C^1$ curve by
the formula
$$
{\rm length}(\gamma) := \int_{H \in {\mathbb {RP}}^{n*}} \#(H \cap \gamma) \, d\mu .
$$
By construction, projective line segments are shortest curves and it can be shown that this definition of length corresponds to a smooth Finsler metric (see~\cite{Szabo:1986} and~\cite{Alvarez-Fernandes:2007}). It follows immediately from the previous formula that if the measure $\mu$ is taken to be equal to the standard $SO(n+1)$-invariant measure everywhere except in the neighborhood of a point $H$ in ${\mathbb {RP}}^{n*}$, we obtain a Finsler metric that agrees with the standard metric outside a neighborhood of the projective hyperplane in ${\mathbb {RP}}^n$ defined by $H$. These metrics can be lifted to the sphere $S^n$ and, indeed, every Finsler metric on $S^n$ for which great circles are geodesics is such a lift. Note that the space of metrics defined by this construction is parametrized by a smooth function---the density for the measure $\mu$---of $n$ variables.

\medskip
\noindent
{\it Circular metrics on ${\mathbb {CP}}^n$ and ${\mathbb {HP}}^n$.} In~\cite{Alvarez-Duran:2002} \'Alvarez Paiva and
Dur\'an Fernandez introduced Finsler submersions and used them to construct examples of metrics
on ${\mathbb {CP}}^n$ and ${\mathbb {HP}}^n$ where complex and quaternionic projective lines are totally geodesic and all
geodesics are (geometric) circles. This was done by using the idea of Busemann sketched above
to construct projective metrics on $S^{2n+1}$ and $S^{4n + 3}$ that are invariant under the Hopf actions of $S^1$ and $S^3$ on these spaces and then projecting those metrics down to
${\mathbb {CP}}^n$ and ${\mathbb {HP}}^n$ by the Hopf map. Note that in the case of ${\mathbb {CP}}^n$ the space of these metrics
is parametrized by a  smooth function of $2n$ variables (i.e., $2n+1$ for a projective metric on $S^{2n +1}$ minus one variable because of $S^1$ invariance), and in the case of ${\mathbb {HP}}^n$ by a smooth function of $4n$ variables.

\medskip
\noindent
{\it Circular metrics on $S^2$.} In~\cite{Alvarez-Berk:2010} \'Alvarez Paiva and Berck constructed
all reversible Finsler metrics on the two-sphere for which geodesics are geometric circles. The construction depends on two functions of two variables: one function to define the circular path geometry and another to metrize it.

\medskip
\noindent
{\it Smooth projective planes.} A smooth real projective plane is a reversible path geometry on ${\mathbb {RP}}^2$ such that any two distinct points determine a unique path and any two paths intersect in only one point. Moreover, the path joining two distinct points and the intersection point of two distinct paths are required to depend smoothly on the points and paths, respectively.

In order to obtain a great number of examples of these planes consider the point-line incidence relation
$$
I := \{(p,l) \in {\mathbb {RP}}^2 \times {\mathbb {RP}}^{2*} : \textrm{ the point $p$ lies on the line $l$ } \} .
$$
As remarked by McKay (Corollary~1 in~\cite{McKay:2005}), it follows from the work of B\"odi and Immervoll~(\cite{Bodi-Immervoll:2000}) that any sufficently small $C^1$ perturbation $\Tilde{I}$ of the point-line incidence relation as a submanifold of ${\mathbb {RP}}^2 \times {\mathbb {RP}}^{2*}$ defines a smooth projective plane:  the path in ${\mathbb {RP}}^2$ corresponding to a point $l \in {\mathbb {RP}}^{2*}$ is the
set of points $p \in {\mathbb {RP}}^2$ such that $(p,l) \in \Tilde{I}$.

There are even examples of smooth projective planes, due to Segre and Immervoll (see~\cite{Immervoll:2003}) that are real analytic. By Theorem~2.2 in~\cite{Alvarez-Berk:2010} every smooth projective plane can be metrized  as smooth reversible Finsler manifolds.

After having reviewed all known constructions of non-Riemannian reversible Zoll Finsler metrics,
we are now ready to pass to the main results of this section.

\begin{maintheorem}{II}
If $(M,F)$ is a smooth, geodesically-reversible Zoll Finsler manifold, then $F$ is the sum of a reversible Zoll Finsler metric and the differential of a smooth function on $M$.
\end{maintheorem}

Before going into the proof of this result, we need a couple of simple lemmas. The first is
a direct consequence of the results of Boothby and Wang~\cite{Boothby-Wang:1958} on regular contact manifolds (see~\cite{Weinstein:1974}  or Proposition~3.3 in \cite{Alvarez-Balacheff:2014}).

\begin{lemma}\label{Zoll-volume}
The volume of an $n$-dimensional Zoll Finsler manifold is equal to an integer multiple of
$(\varepsilon_n n!)^{-1}$ times the $n$-th power of the length of (any) one of its prime geodesics.
\end{lemma}

The second lemma was worked out in a MathOverflow exchange (see~\cite{Alvarez-MathOverflow:2013}).

\begin{lemma}\label{Fundamental-group-Zoll}
The order of the fundamental group of a geodesically reversible Zoll Finsler manifold $(M,F)$  is at most two. In particular, every closed differential $1$-form on $M$ is exact.
\end{lemma}

\proof
 Assume $M$ is not simply connected and let $\sigma \subset M$ be a non-null-homotopic closed curve. By a classic theorem of Hilbert $\sigma$ is homotopic to a closed geodesic and that geodesic is covered a certain number of times $n \in {\mathbb Z}$ by a prime geodesic $\gamma$. Remark now that any two prime geodesics $\gamma_1$ and $\gamma_2$ on a Zoll manifold are homotopic: if $v_1$ is a velocity vector of $\gamma_1$ and $v_2$ is a velocity vector of $\gamma_2$ any continuous path $v_t$ in ${TM \! \setminus \! \mathbf{0}}$ from $v_1$ to $v_2$ defines a homotopy between $\gamma_1$ and $\gamma_2$ by associating to $v_t$ the unique prime geodesic tangent to $v_t$. This shows that the fundamental group of $M$ has only one generator. Moreover, since $(M,F)$ is geodesically reversible, the
 reverse of any prime geodesic $\gamma$ is also a prime geodesic, which means that $\gamma$ is homotopic to its reverse and hence the fundamental group of $M$ has order two.
\qed

\proof[Proof of Theorem~\ref{Zoll}]
Note that since $F$ is geodesically reversible, the reverse metric $F_{-}$ has the same unparametrized geodesics. Moreover, its prime geodesics have the same length as those of $F$.
It follows that the convex combination  $(1-t)F + tF_{-}$ is a continuous path of Zoll metrics
whose prime geodesics have the same length. By Lemma~\ref{Zoll-volume} the volume of all
these metrics must be the same.  However, note that for $t = 1/2$ we have the central symmetrization
$F^S$ whose volume, by Proposition~\ref{central-symmetrization}, must be greater than or equal to that of $F$ with equality only if $F = F_S + \beta$, where $\beta$ is a $1$-form. By Corollary~\ref{adding-form}, $\beta$ must be closed and Lemma~\ref{Fundamental-group-Zoll} implies it must be exact.
\qed

Using Theorem~\ref{Zoll} we can now settle Hilbert's fourth problem for asymmetric metrics on the $n$-sphere.

\begin{maintheorem}{I}
 If $d$ is a continuous asymmetric metric on the $n$-sphere $S^n$ for which all great circles are geodesics, then there exists a (continuous) function $f : S^n \rightarrow {\mathbb R}$ such that
$d(x,y) - d(y,x) = f(y) - f(x)$.
\end{maintheorem}

\proof
By a result of Pogorelov~\cite{Pogorelov:1978} (see also~\cite{Szabo:1986}) every continuous
metric $d$ on the $n$-sphere for which all great circles are geodesics is an uniform limit of distance functions $d_n$ induced by smooth projective Finsler metrics. In the cited papers the
result is stated for reversible metrics, but this hypothesis is never used in the proof.

By Theorem~\ref{Zoll}, for each $d_n$ we have $d_n(x,y) - d_n(y,x) = f_n(y) - f_n(x)$ for some smooth function $f_n$ on the $n$-sphere. Since the functions $f_n$ are defined up to an additive constant, we may fix an arbitrary point $\zeta$ on $S^n$ and assume that all the $f_n$ vanish at that point. Define $f$ as the limit
$$
f(x) :=  \lim_{n \rightarrow \infty} \left( d_n(\zeta,x) - d_n(x,\zeta) \right) ,
$$
which, on account of the uniform convergence of the $d_n$, is a continuous function. It follows that
\begin{align*}
 d(x,y) - d(y,x) &= \lim_{n \rightarrow \infty} \left( d_n(x,y) - d_n(y,x) \right) =
 \lim_{n \rightarrow \infty} \left( f_n(y) - f_n(x)\right) \\
 &= f(y) - f(x) .
\end{align*}
\qed

\section{Finsler metrics without non-trivial integrals of motion} \label{geodesic-rigidity}
Every continuous function on the space of geodesics of a Zoll manifold $(M,F)$ is an integral of motion for the geodesic flow of the metric $F$. At the other end of the spectrum we have Finsler manifolds where the geodesic flow is Anosov or transitive and the only integrals of motion are trivial: constant multiples of the Hamiltonian of the metric. The reader will find many classes of geometrically meaningful examples in the works of Foulon~\cite{Foulon:1986}, Barbosa Gomes, Ruggeiro and Chimenton~\cite{Barbosa-Ruggiero:2013, Barbosa-Chimenton-Ruggiero:2020}.

In two dimensions, geodesic rigidity theorems for metrics without non-trivial integrals of motion are very easy to prove:  if $F_1$ and $F_2$ have the same geodesics and no non-trivial integrals of motion, then by Proposition~\ref{integral-of-motion} the differentials of the Hilbert $1$-forms are
multiples and therefore one metric is a multiple of the other plus a closed $1$-form. The main problem in $n$ dimensions ($n > 2$) is to extract meaningful geometric information from the proportionality of $d\alpha_{F_1}^{n-1}$ and $d\alpha_{F_2}^{n-1}$. This difficulty is solved
by a simple application of the theory of areas in Finsler spaces.

\begin{theorem} \label{areas}
Let $F_1$ and $F_2$ be two $C^2$ Finsler metrics defined on a manifold $M$ which have the same unparametrized geodesics. If the geodesic flow of either metric does not admit non-trivial
integrals of motion, there is a constant $\nu > 0$ such that the Holmes-Thompson area of any hypersurface in $(M,F_2)$ is $\nu$ times its area in  $(M,F_1)$.
\end{theorem}

\proof
Since $F_1$ and $F_2$ have the same unparametrized geodesics, Proposition~\ref{integral-of-motion}
tells us that there exists a positive, continuous function $\nu$ on the manifold of tangent directions $STM$ that is constant along geodesics and such that  $d\alpha_{F_2}^{n-1} = \nu d\alpha_{F_1}^{n-1}$. By hypothesis,  $\nu$ is a constant.

 Let $N \subset M$ be a hypersurface, which, by the locality of area, we can assume to be contained in a region $B \subset M$ that is convex for both $F_1$ and $F_2$. The set of (common) geodesics in $B$ is a manifold $\geod{B}$ and we denote by $\omega_1$ and $\omega_2$ the maximally-non-degenerate, continuous $2$-forms induced by the differentials of the Hilbert forms $\alpha_{F_1}$ and $\alpha_{F_2}$ in $STM$. The equality $d\alpha_{F_2}^{n-1} = \nu d\alpha_{F_1}^{n-1}$ translates to the equality  $\omega_2^{n-1} = \nu \omega_1^{n-1}$, which in turn yields
$$
 \int_{\gamma \in \geod{K}}
\#(\gamma \cap N) |\omega_2^{n-1}| =
\nu \int_{\gamma \in \geod{K}}
\#(\gamma \cap N) |\omega_1^{n-1}| .
$$
By the Crofton formula for hypersurfaces in Finsler spaces (Theorem~\ref{crofton-hypersurface}) the area of $N$ in $(M,F_2)$ equals $\nu$ times the area of $N$ in $(M,F_1)$.
\qed

\begin{maintheorem}{VI}
Let $F_1$ and $F_2$ be two reversible $C^2$ Finsler metrics defined on a manifold $M$ and with the same unparametrized geodesics. If the geodesic flow does not admit any non-trivial integral of motion, then the metric $F_2$ is necessarily a  constant multiple of $F_1$.
\end{maintheorem}

\proof
Reversible Finsler metrics are completely determined by their hypersurface area density (Proposition~\ref{areas-determine-metric}), therefore Theorem~\ref{areas} implies that $F_2$ is a constant multiple of $F_1$.
\qed

\begin{maintheorem}{V}
Let $M$ be a two- or three-dimensional manifold and let $F$ be a geodesically reversible Finsler metric on $M$. If the geodesic flow does not admit any non-trivial integral of motion, then the metric $F$ is necessarily the sum of a reversible metric and a closed 1-form.
\end{maintheorem}

\proof
Since the Finsler metric $F$ is geodesically reversible it has the same un\-parametrized geodesics as its central symmetrization $F^S$. Theorem~\ref{areas} tells us that there exists a constant $\nu > 0$ such that the hypersurface area density of $(M,F^S)$ equals $\nu$ times that of $(M,F)$. By definition, the hypersurface area density of the metric $F$ equals that of its areal symmetrization $F^B$ and, since both $F^S$ and $F^B$ are symmetric, Proposition~\ref{areas-determine-metric} implies that  $F^S$ and $F^B$ are proportional and, a fortiori, conformal.  Proposition~\ref{constant-width-constant-brightness} now implies that $F = F_S + \beta$, where $\beta$ is a $1$-form, which is closed by Proposition~\ref{adding-form}.
\qed

Theorems~\ref{hyperbolic-space-forms-non-reversible} and~\ref{hyperbolic-space-forms-reversible}
follow immediately from the previous results by recalling that the geodesic flow on a hyperbolic space form of finite volume is transitive.

redpajama_set_name: RedPajamaArXiv


View the active MLS listings of Naperville homes for sale. Use the Advanced Search button above to refine your results.
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text:
No 2020 racing season has been as-planned anywhere in the world due to the Coronavirus Pandemic. Whilst The Scottish Championship has been officially cancelled, meaning the team's title defence will have to wait until 2021, there is still track-action to be had!
The team kicked-off 2020 with a launch video and photoshoot, airing in a 'virtual launch' on the 25th of July . These are followed-up by the first on-track action since March, as the team take to the track at Knockhill Racing Circuit for an official test day on the 1st of August. Stay tuned to this website and the team's social media channels for further on-track news as the year progresses!
You can re-watch the team's launch video here and view photos from the launch below (select a photo to view it larger).

redpajama_set_name: RedPajamaCommonCrawl


Home Death Doesn't the Bible phrase "unquenchable fire" indicate that the hell fire never goes out? What about the phrase "eternal fire"?
Doesn't the Bible phrase "unquenchable fire" indicate that the hell fire never goes out? What about the phrase "eternal fire"?
"He will thoroughly purge his floor, and gather his wheat into the garner; but he will burn up the chaff with unquenchable fire." (Matthew 3:12).
"Even as Sodom and Gomorrha, … are set forth for an example, suffering the vengeance of eternal fire." (Jude 1:7). We can see in this verse that Sodom and Gomorrah were destroyed with everlasting fire as an example of hell, but those cities are not still burning. They were turned into ashes (2 Peter 2:6), which is what eternal fire does. It is eternal in its consequences.


text:
Мальвина Зиновьевна Швидлер (19 августа 1919, Одесса — 15 июля 2011, Киев) — советская и украинская актриса театра и кино. Народная артистка Украины (1996).

Биография
Мальвина Зиновьевна Швидлер родилась 19 августа 1919 года в Одессе в семье управляющего кирпичного завода. В 1939 году закончила Одесское театральное училище и играла в Одесском театре миниатюр. В 1939—1941 годах была актрисой Львовской филармонии, во время войны служила в 1941—1944 годах в Ташкентском академическом русском драматическом театре им. М. Горького. После войны с 1945 года была актрисой Киевского государственного театра русской драмы им. Леси Украинки.

Умерла 15 июля 2011 года после продолжительной болезни. Похоронена в Киеве на Байковом кладбище (участок № 1).

Награды
 Медаль «За трудовое отличие» (30.06.1951).
 Заслуженная артистка Украинской ССР (1972).
 Народная артистка Украины (18.01.1996).

Работы в театре

Театр русской драмы имени Леси Украинки

 «Горе от ума» (1945)
 «Дамы и гусары» (1946)
 «Обыкновенный человек» (1946)
 «Каменный властелин» (1946)
 «Памятные встречи» (1946)
 «Бессмертие» (1947)
 «Женитьба Фигаро» (1947)
 «Московский характер» (1948)
 «Госпожа министерша» (1948)
 «Сказка о правде» (1948)
 «Карьера Бекетова» (1949)
 «Обрыв» (1949)
 «У порога» (1949)
 «Директор» (1950)
 «Свадьба с приданым» (1950)
 «Жизнь начинается снова» (1950)
 «Учитель танцев» (1950)
 «Тридцать серебреников» (1951)
 «Рассвет над Москвой» (1951)
 «Слуга двух господ» (1952)
 «Мертвые души» (1952)
 «К новому берегу» (1953)
 «Раки» (1953)
 «Весна в Москве» (1953)
 «Стрекоза» (1954)
 «Мораль пани Дульской» (1955)
 «Три сестры» (1955)
 «Годы странствий» (1955)
 «Ложь на длинных ногах» (1956)
 «Рассвет над морем» (1957)
 «Второе дыхание» (1957)
 «В поисках радости» (1958)
 «Потерянный сын» (1961)
 «Забавный случай» (1960)
 «Океан» (1961)
 «Перед ужином» (1962)
 «Центр нападения умрёт на рассвете» (1962)
 «Лес» (1962)
 «Нас где-то ждут» (1963)
 «Сенсация № 1» (1965)
 «В день свадьбы» (1964)
 «Дон Карлос» (1965)
 «Хождение по мукам» (1966)
 «Встречи поздние и ранние» (1966)
 «Брак по конкурсу» (1967)
 «Далекие окна» (1967)
 «Традиционный сбор» (1967)
 «На всякого мудреца довольно простоты» (1969)
 «Странная миссис Сэвидж» (1968)
 «Правду! Ничего, кроме правды!!!» (1969)
 «Разгром» (1970)
 «Мария» (1970)
 «Дети Ванюшина» (1970)
 «Великоманиев» (1971)
 «Затюканный апостол» (1971)
 «Каменный властелин» (1971)
 «Бесприданница» (1973)
 «Второе свидание» (1972)
 «Добряки» (1973)
 «Последние дни» (1974)
 «Власть тьмы» (1974)
 «Странный доктор» (1975)
 «Без труб и барабанов» (1975)
 «Как важно быть серьёзным» (1976)
 «Кремлёвские куранты» (1977)
 «Хозяйка» (1978)
 «Пять дней в июле» (1980)
 «Игрок» (1982)
 «Филумена Мартурано» (1983)
 «Бумажный патефон (Счастье мое)» (1984)
 «ОБЭЖ» (1985)
 «Мамаша Кураж и её дети» (1986)
 «Священные чудовища» (1987)
 «Кровавая свадьба» (1988)
 «Перламутровая Зинаида» (1988)
 «Самоубийца» (1989)
 «Без вины виноватые» (1991)
 «Метеор» (1992)
 «Генералы в юбках» (1993)
 «Двери хлопают» (1996)
 «И всё это было… и всё это будет…» (2001)

Фильмография
  — Третий удар — татарка
  — Приключения с пиджаком Тарапуньки (короткометражный фильм) — швея
  — Гадюка — Роза Абрамовна, соседка Ольги Зотовой по коммуналке
  — Наедине с ночью
  — Путь к сердцу — врач-ординатор
  — Где вы, рыцари? — мама Жанночки Ситниковой
  — Хозяйка
  — Имя твоё — эпизод
  — Западня — ''Ида, служанка у Вагмана

Озвучивание мультфильмов
  — Какого рожна хочется?
  — Алиса в Стране чудес
  — Савушкин, который не верил в чудеса

Примечания

Ссылки
 «Бульвар Гордона»: Интервью с актрисой.
 Любовь Журавлева «Зеркало недели» И смех, и слезы — и любовь. Актриса Мальвина Швидлер: разрозненные страницы
 19 августа 1919 года в Одессе родилась Мальвина Зиновьевна ШВИДЛЕР — актриса театра и кино. Народная артистка Украины (1996).
 Мальвина Швидлер. Жизнь как детская рубашка... (2013)

Персоналии:Киевский русский драматический театр имени Леси Украинки
Похороненные на Байковом кладбище

redpajama_set_name: RedPajamaWikipedia


text:
Agfa promotes individuality among new corporate members
Managers resist urge to impose old bureaucracy Changing the name of Agfa's medical imaging business last year to Agfa HealthCare was more than a cosmetic move. The unit, which traditionally focused on imaging, is undergoing a
Managers resist urge to impose old bureaucracy
Changing the name of Agfa's medical imaging business last year to Agfa HealthCare was more than a cosmetic move. The unit, which traditionally focused on imaging, is undergoing a metamorphosis, broadening its customer base, modifying distribution channels, and repackaging products.
Driving this change is an aggressive strategy aimed at boosting revenues and improving the company's position within the healthcare IT marketplace. The recent acquisition of long-time Agfa ally Mitra and a minority equity stake in MediVision Medical Imaging reflect this strategy.
Mitra now provides in-house expertise in critical software development. MediVision, a developer of digital imaging devices for ophthalmic applications, exemplifies Agfa's hunger for expansion into complementary healthcare arenas. Following acquisition of Talk Technology, a creator of speech recognition tools, and medical patient record developer Quadrat, strategists began to position Agfa Healthcare as a key provider of informatics products that extend well beyond the traditional bounds of radiology.
To get to this point, however, Agfa has created as many challenges as it has opportunities. The acquisitions bring a need for integration into the much larger whole of Agfa, which marshals commercial initiatives in far-flung ventures including nondestructive testing, motion pictures, and photography. Bureaucracies that rule such varied businesses often create generic governance that can stifle creativity. John Glass, general manager of Agfa HealthCare, argues that his unit will not fall prey to such an approach. Agfa HealthCare, he said, can determine its own fate because its sales account for most of Agfa's gross revenues. Glass' election last fall to the Agfa-Gevaert board of directors bears witness to the importance of the healthcare business unit.
"Within our company, we are the favored son," Glass said. "We get a lot of attention from the board and from investors."
To protect this status and the autonomy that comes with it, Glass and his colleagues have constructed a fire wall between informatics, exemplified by Mitra, and the general Agfa bureaucracy.
"The integration (of this company) will not be to the point where the Agfa culture is dominant," he said. "We will maintain the Œsoftware' culture and not put our own shadow on that."
Glass characterizes this culture as "California casual"‹staff wear colored shirts with company logos and meet for barbecues, cubicles are the same size, regardless of rank, and employees have a vested interest in their company's success. Agfa owns 93% of Mitra, and employees own the rest.
To help insulate Mitra from bureaucratic forces that might slow down progress, Glass wins board approval for the concepts behind informatics projects, then works the operational issues, such as interim approvals based on milestones, among his own staff within the HealthCare unit.
Preserving the native environment of acquired companies, especially those of software developers, makes sense intuitively. But whether it breeds continued productivity is hard to say. Agfa managers know how to measure productivity in film and equipment manufacturing but not software development, Glass said.
"I can't tell you that this (environment) enhances productivity," he said. "But it has always been there, and we need to have some benchmarks."
Coping with unique circumstances is part of the challenge of making the transition from a business focused on medical imaging to one that addresses the broader venue of healthcare. Although Agfa's roots extend to photography, as do those of competitors such as Kodak and Konica, Agfa has tried to tap other resources‹other core technologies‹that will promote the long-range goal of integrating information islands within medical practice.
Through acquisitions and internal development, the company has assembled a product portfolio that spans PACS, speech recognition, hospital and radiology information systems, and the electronic patient record. Getting all these pieces to work together is an integration challenge as well as a prospecting one. Agfa managers need to find other, still missing pieces to make all the right connections. Acquisitions, therefore, will continue to play an important role in the company's growth strategy. But the process is laborious and time-consuming, Glass said.
"For every (prospect) that comes along, we have to kick the tires," Glass said. "We have been kicking a lot of tires."
In the meantime, integrating the pieces already in hand is a high priority. More is involved than just bringing the new companies into the corporate fold. Strategic plans and tactics must be adjusted to make the most of opportunities. Horizontal expansion requires corporate exploration into unfamiliar ground such as pathology, ophthalmology, cardiology, and orthopedics. These forays require solutions to unique workflow problems. The company is also looking to move vertically, to tie together operations within and among various medical departments.
Increased productivity by medical practitioners is a primary objective. This might be achieved by forging links among "medical content areas," said Robert E. Cooke, global general manager of IMPAX Solutions. The company has created, for example, a "PACS watch" subsystem that gathers statistics and compares them with benchmarks, allowing the customer to make informed decisions about modifying workflow. Another example is the development of productivity tools that are sensitive to the content within images. Agfa engineers have developed an electronic atlas of the body to replace the hard-copy reference that commonly accompanies diagnostic workstations.
"Physicians interpreting exams can pull up content dynamically and, in the context of the image being viewed, gain additional information that may be useful in differential diagnosis, planning treatments, or gaining CME credits," Cooke said.
These products need to be effectively launched into the market. Agfa must leverage and modify existing distribution channels to fit these new needs. Distribution channels designed to address academic institutions will be used to address group purchasing organizations. In adjusting the distribution channels, new opportunities will arise to sell products that might not otherwise have been created. A single scaleable system, for example, might give rise to several other products tailored to specific needs and price points.
"In this way, we put the right products into the right distribution channels so we can be sure we are touching all the markets and leveraging our resources appropriately," said Douglas M. Tucker, Ph.D., vice president of IMPAX Solutions.
The Agfa strategy accounts for threats as well as opportunities. These threats may be competitive, economic, or technological, Glass said. Ink jet printing addresses all three forms of threat. Institutions looking to get more for less in economically challenging times will be tempted to buy into this technology, which promises cost savings in both its acquisition and its operation. Ink jet technology is not yet accepted as part of a diagnostic solution, but that day is coming, Glass said.
"We can already demonstrate, through our photo division, color ink jet technology that delivers hard copy that is very hard to distinguish from photographic prints," Glass said. "We have to assume that it can be done eventually in diagnostics."
Agfa has a development program under way to create high-resolution ink jet printers. But when medical ink jet printing arrives, the competitive landscape of the industry will change, he said. New companies, ones based predominantly in Japan, will be jockeying for position in this emerging segment of the market. And Agfa will have to battle a whole new wave of competitors.
As with ink jet printing, informatics presents competitive challenges. Managers must read the market carefully, recognize the potential of technology, and respond to threats in a timely fashion.
"I think the biggest threat is making sure we invent the future fast enough," Cooke said. "To do that we need to be efficient; we have to react quickly to trends in the market and trends in technology."
Under some circumstances, that will mean developing new technologies. Or it may mean stopping activities that will no longer provide a good return on investment.
"We have to be prepared to make tough decisions‹to stop activities that don't make sense in a new environment and to start activities that do, because you can't do everything," he said.


The Albin Scampi is a 29'10" monohull sailboat designed by Peter Norlin. She was built by Albin Marine (Sweden) and made of GRP (glass reinforced polyester). The production started in 1979 .
The Albin Scampi belongs to the Half Tonner class.


and the stories behind them. I have just watched the Queen film, Bohemian Rhapsody, and was amazed by everything! There are some factual exaggerations – including the fact Queen did not sell their tour van to fund their debut album; Freddie Mercury did not reveal his HIV/AIDS diagnosis to the band just before Live Aid in 1985 – but the performances are exceptional and it is great to get a closer look at the legendary band. One of the factual errors arrived when the band were presenting Bohemian Rhapsody to the record label boss, Ray Foster (played by Mike Myers in the film). There are errors and oversights in the film but it is not a documentary – they had to exaggerate some of the truth! The band's greatest song, Bohemian Rhapsody, must have seemed like a comet from space in 1975. We have not heard anything like it since and it is a song that could only have come from the brain of Freddie Mercury. It is considered one of the finest songs ever recorded and certainly cracks my top-five. I love how it three songs squashed into one and is this quasi-operatic expression. It is one of those songs that should be a benchmark for any ambitious songwriter who does not care about limits, being commercial and following the pack.
When it was unveiled to the world, it got some average reviews and very little airplay given its six-minute-plus running time. D.J. Kenny Everitt did play it (numerous times) but very few others would touch it! Things have not really changed a lot now and I wonder whether Bohemian Rhapsody would get played on a big station in 2019. Bohemian Rhapsody did go to sell more than a million copies by the end of January 1976 and stayed at the top of the U.K. charts for nine weeks. It was re-released in 1991 following Mercury's death and it remains one of the best-selling songs ever. Mercury wrote the song at his London home and one can only imagine what was racing through his mind as he started putting it together. Roy Thomas Baker – who co-produced the song alongside Queen – has said how Mercury played the opening ballad section and then, at a point, stopped to say this is where the Opera comes in! Imagine a songwriter doing that today and it would be met with stony faces. It is one of those songs that was in Mercury's mind and almost complete before he took it to the band. A lot of Queens songs required a lot of studio time and attention but, in terms of effort and studio demands, Bohemian Rhapsody tops them all! Many have debated what the title refers to but there is that combination of a rhapsody and operatic drama combined with the bohemian artist being an individual.
It sounds like a contradiction in terms – if that is the right term?! – but there is a lot of speculation as to what was in Mercury's mind. I cannot recall whether there has been a song since 1975 that has contained so much ambition, different turns and twists and a big plot – many wonder, again, what the song is about and whether Freddie Mercury was borrowing from his own life. Although the song was completed in 1975, Mercury started writing the lyrics in the late-1960s. Lyrics from older songs were snipped and put into Bohemian Rhapsody. When the band started recording at Rockfield Studio 1 in Wales, they could not have imagined what they would produce and how it would be received. There are rumours Mercury had the song completed in his head before entering the studio and, like a conductor, guided the players and knew exactly how it was going to sound. Mercury, Brian May and drummer Roger Taylor sang their vocal parts for ten-twelve hours a day and it shows how intense things were! Can anyone imagine a modern-day band doing that?! Mercury was determined to get the sound he heard in his head so there was a lot of vocal work! He was very much in control because he wanted it to be as magnificent and grand as possible. One imagines there would have been some tense moments in the studio but all the hard work and hours led to one of the greatest songs ever.
Bohemian Rhapsody confused many at the time because it does not have a conventional structure and narrative. The song has no chorus and it was rare to hear a band stray away from love to address subjects as dark as murder and nihilism. From the narrator telling his mother than he'd just killed a man – putting a gun against his head – to the biblical guitar solo that would come, Bohemian Rhapsody is this extraordinary masterpiece that one can endlessly pull apart. Maybe Mercury was taking from his own regrets and experiences. Whether he was talking about confusion and changes or merely penning something fictional, the lyrics are certainly different and very interesting. There is the casual disregard for life – Mercury explaining how nothing really matters anymore – and this very open and honest confession. Before you get attached to this ballad and familiar sound, the song goes semi-operatic and there are a series of harmonic changes. Think of how The Beatles pushed the studio with 1967's A Day in the Life and The Beach Boys' Good Vibrations was a revelation; Bohemian Rhapsody was unheard of and it must have been an immense task getting it all laid down! Whether you prefer the fatalistic lyrics, the wall of vocals or the wonderful twists throughout, it is hard to beat the head-banging solo. It was immortalised for Wayne's World and is one of the most distinctive passages in Rock history.
Is it possible to reach that crescendo and release without letting it all go and submitting to the song's primal and fantastic energy?! In terms of the music and structure, it was hugely progressive and would be imitated by endless Psychedelic-Rock bands. I think Bohemian Rhapsody will never be bettered in terms of its ambition because, now, we want songs to be punchy and many people do not have the patience to sit through a song that is over six minutes in length. Many have studied the lyrics and feel it is Freddie Mercury unveiling childhood fears and harsh memories. Some say the song is straight and direct whereas many others have gone in different directions – will we ever truly know the secrets and truths behind Bohemian Rhapsody?! There are reports Mercury dismissed the song as rhyming nonsense with no personal depth but maybe that was him putting people off of the scent. Musicians, critics and fans heaped praise on Bohemian Rhapsody's complexities and genius – Brian Wilson was especially blown away and moved. Progressive-Rock definitely had its time and place – before Punk made it unfashionable – and Bohemian Rhapsody must be considered the king of all the genre's songs. So many other bands laced together strands of songs but nothing touched the majesty and wonder of Bohemian Rhapsody. Maybe Stairway to Heaven from Led Zeppelin – released in 1971 – has that three-part stricture but is nothing like Bohemian Rhapsody.
FEATURE: Immune to the Muse: Why Are Women Not Being Booked as Festival Headliners?


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text:
Thread: Pick your wins and losses
OL FU
Proud Member of the AGS Wine and Cheese Crowd
Time to step up and prove your powers of prognostication or just how much of a homer you really are.
I think we have done this before (we have talked about everything else related to football so I don't think I am going out on a limb) so why not again. I have to admit I am not quite as confident as I may have been in other years. By many other teams' measurements, FU had a good year last year and even considering our 8-3 record we did ok. But man we were shaky. 8-3 with lots of close wins. Of course, we blame the injuries and I will take that argument so assuming we don't have any critical ones here goes:
Sept 1 – Presbyterian College
Welcome to FCS The last time we played was Ingle Martin's debut as a Paladin and even though ESPN thought we lost it wasn't even close. The time before that was 2001 (and we had a pretty good team that year) and PC played a whale of a first half with a dynamite QB and offense. They scored 28 in the first 30 minutes but lost by 20 at the end of the game. This year should be interesting and I expect them to play close (during the first half) as we play it safe. But we win by three scores. 35-14.
September 8 - @ Hofstra
The last time we played was 2005 and Furman won after about 15 overtimes. Hofstra passed and we watched them pass. Last year was not the same Hofstra team. They played everyone close and lost to everyone. I really have no idea so I will have to go with what I have been hearing from the CAA fans and they don't seem to think Hofstra is going to win a whole lot more than last year. I wish I was that confident. Tough to play in front of 500 rabid fans. Furman wins 28 -24.
Sept 15 – @ Clempscum
I don't dislike them nearly as much as I act on this board. The last time we played we lost 17-28 and surprisingly outscored them in the second half. Take away a couple of fluke plays and the game is even closer. But it was the start of the end of Clemson's pass happy offense. They were determined to run and they did. They played vanilla and we kept it close. I am not optimistic about this year. The last time we beat the Tiggers FDR was president. Clemson 35-10.
Sept 29 – Wofford
If a miracle occurs the week before and the Terriers upset the Mountaineers, they will be flying high and if they don't, they will realize that their special season could be ending early. We have had some good games recently. 2003 is the most recent Wofford win and you have to go back more than a little to find the next most recent. That game was pretty close considering Wofford was on their way to a SoCon Championship and FU was mired in mediocrity. 2004 and 2005 were close but FU wins. Last Year FU ran all over them but it was one of our few healthy games and one where Wofford was not healthy. I do think Wofford is going to be very good but not good enough. Furman 28-21.
Oct 6 Coastal Carolina.
Most of us FU fans were shocked and disappointed last year and we should not have been. All we needed to do was look at CCU's home record and who they had beat before in Conway to realize that this was an upset in the making. But this ain't in Conway and while I don't think CCU will be as down as a lot of people believe, we will be better and they won't be. Furman 34 CCU 21
Oct 13 @ The Citadel
I have to admit that my confidence level on this pick is not great. Playing in Charleston is never easy. The Citadel cadets live and die by their honor code (of course, it does not include class and dignity when confronted by the enemy ). Great Coach, good team coming back and they will play like there is no tomorrow. It would not surprise me if we see overtime again in this one. But Furman wins 28-24.
Oct 20 @ Chattanooga
Every year it is the same, " Is this the year that Chattanooga comes back?" If you pick them every year for the next 20 years you are bound to hit it right sooner or later . So is this the year? Could be and if you beat us plus a few others you can shake your finger in my face and say you told me so. I will believe it when I see it. A healthy Furman wins 24-10.
Oct 27 Appalachian State – Probably the longest history of close games in the I-AA history of the SoCon. Home field plays a huge advantage and ASU has to lose somewhere so it might as well be here. Furman 27-24.
I will admit to a slight amount of homerism here.
November 3 Elon
Elon improved last year. But the loss of their QB makes one go . After a fairly good season last year they ended the year horribly. I wonder (totally without fact) if some of the Lembo problems that we heard so much from the Lehigh fans hasn't found its way into Elon. Time will tell. Furman is off after ASU but wins 21-10.
November 10 Georgia Southern
By this time the Igglets have found their grove, and they will either be playing to not have their fourth loss (or to gain or playoff bid) or playing to beat the team they most love to beat. This time of year is a good time to buy stock in drug companies as purple induces extremely high blood pressure in gnat infested south Georgia. I hate to say it, but we don't fair well when the eagles are revved up and they will be. GSU wins 24-21.
November 7 Western Carolina
Last game of the season, Furman has had their FU last game and are now playing for seeding. Western Carolina just got their tails kicked the week before by a mad ASU team and now they are playing for pride against a team playing for home field in the playoffs.
Furman 35 -17.
Honestly, I think 9-2 is doable but is the best that can be expected. We play a lot of away games this year. 8-3 is more likely but it is hard to choose who might deliver the the other loss. The honest truth it would be just as easy to have said we beat GSU and lose to ASU and the Citadel (Sorry Wofford, not ready to give you one on our home field)
Even 7-4 is possible but would be extremely disappointing with an experienced team coming back. for not injuries
I am sticking with 9-2 and if we lose 3 then I will change my picks
Anti-homerism: pick your most-likely LOSSES of the season!
By Screamin_Eagle174 in forum FCS Discussion
Last Post: August 6th, 2008, 01:12 PM
Predict NAU's number of wins and losses this year
By Jacks76 in forum FCS Discussion
Last Post: June 10th, 2007, 04:49 PM


text:
Q: Logical negation of the Brocard's conjecture Let $p_{k}$ and $p_{k+1}$ are consecutive prime numbers greater than $2$.
The Brocard's conjecture states that:
Conjecture: For all $k≥2$, there are at least four prime numbers between $p^2_{k}$ and $p^2_{k+1}$.
My question is: How one can logically negate this conjecture.

A: To approach this problem, let us first translate the question into a statement in first-order logic. This is the usual language which one uses when considering mathematical statements. If we let the relation $P(k)$ stand for the statement there are at least 4 prime numbers between $p_k^2$ and$p_{k+1}^2$ and let $T(k)$ represent the statement that $k\geq 2$, then the statement can be represented by the following first-order formula where $\to$ represents the logical connective "implies":
$$\forall k(T(k)\to P(k))$$
Now, once we have translated this conjecture into a first-order formula, we can apply the mechanical rules of first-order logic to negate the statement using the following logical equivalences where $\leftrightarrow$ is the logical connective "if, and only if" and can be thought of as meaning "is equal to."
If $\phi$ and $\psi$ are formulas in a first-order language, then:

*

*$\lnot\forall x\phi\leftrightarrow \exists x\lnot\phi$

*$\lnot (\phi\to\psi)\leftrightarrow (\phi\land\lnot\psi)$
Applying the above two rules to our first-order formula $\forall k(T(k)\to P(k))$ we obtain:
\begin{align*}
\lnot\forall k(T(k)\to P(k))&\leftrightarrow \exists k\lnot(T(k)\to P(k))\\
&\leftrightarrow \exists k (T(k)\land \lnot P(k)).
\end{align*}
Therefore, we find that the negation of our formula $\forall k(T(k)\to P(k))$ is logically equivalent to $\exists k (T(k)\land \lnot P(k))$ and we can translate this statement back into English using our interpretations of the relations $T(k)$ and $P(k)$ as, "There exists some $k$ where $k\geq 2$ and where it is not true that there are at least 4 prime numbers between $p_k^2$ and$p_{k+1}^2$." I trust that you realize that the negation of, "there are not at least four prime numbers between $p_k^2$ and$p_{k+1}^2$" is equivalent to saying that, "there are less then four prime numbers between $p_k^2$ and$p_{k+1}^2$," and, using this, our statement:
"There exists some $k$ where $k\geq 2$ and where it is not true that there are at least 4 prime numbers between $p_k^2$ and$p_{k+1}^2$."
can be simplified to the more natural:
"There is some $k\geq 2$ where there are less than four primes between $p_k^2$ and$p_{k+1}^2$."
As you can see, the hard part is effectively translating our statement into a first-order language. Once this is done, we can apply the mechanical rules of first-order logic to find an appropriate formulation for the negation of the statement, which can then be re-translated into English.

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Ron and Sonny in 1994 holding the final wood model of the PV-3.
The top and underside of first concept model of the Kozy Kollar . The frame was made using copper pipe, what else would a couple of plumbers use.
A metal frame is added to the third Kozy Kollar development.
The fourth version has a arched end to allow for the Diverting gasket to be fitted. The seal was not perfected yet but we're gettng close.
This wooden model now has the Diverting Seal that will go along with the Main Seal and assure that the Kozy Kollar will work every time.
A 2017 photo of Ron and Sonny holding the PV-3. Looks like the Kozy Kollar aged better then we did!
The Fish Tank Display was used at counter days and trade shows. The tank has Lexan sides and bottom and was mounted on a plywood box.
The inside of the tank was shingled and a Kozy Kollar was installed. We would fill it with 4 to 5 gallons of water to show how well the Kollar sealed the shingled tank bottom and around the pipe.
A local supply house would let us use the Fish Tank as a counter display for extended periods of time.
Apparently a mischievous plumber decided to have some fun and slid the pipe up so it was just inside the Koller. It did not take long before a curious customer reached in the tank to play with the pipe.
With just the slightest movement the pipe came out and 5 gallons of water poured out soaking the customer, flooding the counter and floor! I don't think the tank finished draining before we received the call telling us to " Get that thing outa here". The Fish Tank has since retired from public life.
The Rain Display ran water down the roof section to show how the Diverting Seal directs most of the water around the Kollar.
This display was used when smoking was allowed indoors, funny it doesn't seem that long ago. At an Indiana PHCC show we noticed people stopping for a few seconds to look at the display which is exactly what we hoped they would do.
At some point we realized they weren't as interested in our product as we thought they were. People were using the display as an ash tray! The display was full of cigarette butts!
At Kozy Kollar we try to do everything we can to keep the cost down. The Rain Display is a perfect example.
Using scrap aluminum angle, a used Taco circulator, old tank stand, shower drain, copper, and PVC pipe and fittings we put this display together for less then $30.
We truly are a couple of frugal N.H. Yankees!


text:
26315 open access dissertations and theses found for:
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Full Text - PDF (1,001.95 KB)
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Principals' Understanding of Teacher Evaluations Connected to the Colorado Student Assessment Program
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