  Let $M$ be a compact connected oriented $n-1$ dimensional manifold without
boundary. In this work, shape space is the orbifold of unparametrized
immersions from $M$ to $\mathbb R^n$. The results of \cite{Michor118}, where
mean curvature weighted metrics were studied, suggest incorporating Gau{\ss}
curvature weights in the definition of the metric. This leads us to study
metrics on shape space that are induced by metrics on the space of immersions
of the form $$ G_f(h,k) = \int_{M} \Phi . \bar g(h, k) \vol(f^*\bar{g}).$$ Here
$f \in \Imm(M,\R^n)$ is an immersion of $M$ into $\R^n$ and $h,k\in
C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$. $\bar g$ is the standard
metric on $\mathbb R^n$, $f^*\bar g$ is the induced metric on $M$,
$\vol(f^*\bar g)$ is the induced volume density and $\Phi$ is a suitable smooth
function depending on the mean curvature and Gau{\ss} curvature. For these
metrics we compute the geodesic equations both on the space of immersions and
on shape space and the conserved momenta arising from the obvious symmetries.
Numerical experiments illustrate the behavior of these metrics.

Differential Geometry