Question: I am reading John Lee's Riemannian Geometry Chapter 3, and I want to do some exercises. I think that I need some hints to solve the following: (Problem 3-8 of that book)

Suppose $M$ and $N$ are smooth manifolds, and $p: N \to M$ is a surjective submersion. If $N$ has a Riemannian metric $h$, at each point $x \in N$ the tangent space $T\_x N$ decomposes into an orthogonal direct sum $T\_x N = H\_x \oplus V\_x$, where $V\_x := Kerp\_∗ = T\_x N\_{p(x)}$ is the vertical space and $H\_x := V\_x^ {\perp} $ is the horizontal space. If $g$ is a Riemannian metric on $M$, $p$ is said to be a Riemannian submersion if $h(X,Y)=g(p\_∗X,p\_∗Y )$ whenever $X$ and $Y$ are horizontal. 

(a) I want to Show that any vector ﬁeld $W$ on $N$ can be written uniquely as $W = W^H + W^V$ , where $W^H$ is horizontal, $W^V$ is vertical, and both $W^H$ and $W^V$ are smooth.

(b) secondly i have to show If $X$ is a vector ﬁeld on $M$, show there is a unique smooth horizontal vector ﬁeld $Z$ on $N$, called the horizontal lift of $X$, that is $p$-related to $X$. (This means $p\_∗Z\_q = X\_{p(q)}$ for each $q \in N$.)

Answer: 
Let $n,m$ be the dimension of $N,M$ respectively, and let $n-m=k$.

(a) Given a vector field $W$ on $N$, it is obvious that there is a unique way to decompose $W$ to a sum of a vertical field and a horizontal one. The point is to show that these vector fields turn out to be *smooth*. It suffices to show that around any $x\in N$ there is a smooth frame of the tangent bundle, $X\_1,\ldots,X\_n$, such that $X\_1,\ldots,X\_k$ is an orthonormal frame of the vertical tangent bundle, and $X\_{k+1},\ldots,X\_n$ is an orthonormal frame of the horizontal bundle (then it follows immediately that the orthogonal projection at each point, $\pi\_x:T\_xN\to V\_x$, changes smoothly and this is all we need).

Let $x\in N$. We know from the inverse function theorem that there is a coordinate chart around $x$, $\varphi:U\subset N\to\mathbb{R}^n$, such that $d\varphi$ maps the vertical space at any point in $U$ to $\{(x\_1,\ldots,x\_k,0,\ldots,0)\}\subset\mathbb{R}^n$. Let $\partial\_{x\_1},\ldots,\partial\_{x\_n}$ denote the vector fields that are induced by $\varphi$, hence $\partial\_{x\_1},\ldots,\partial\_{x\_k}$ span the vertical bundle. Our desired smooth orthonormal frames can now be obtained by performing the Gram Schmidt algorithm simultaneously along $U$ (the frames we get will indeed be smooth since the metric is smooth).

(b) Note that for any $x\in N$, the restriction $dp\_x|\_{H\_x}:H\_x\to T\_{p(x)}M$ is an isomorphism hence again, it is clear that a vector field on $M$ can be lifted to a unique horizontal vector field on $N$, and we only need to show smoothness.

Let $U\subset N$ an open subset, and let $X\_{k+1},\ldots,X\_n$ be a smooth orthonormal frame of the horizontal bundle along $U$. $dp|\_{H\_x}$ is smooth, and maps each of the $X\_i$'s to a smooth vector field on $M$. Since taking inverse is a smooth function on the space of invertible linear maps (or the space of invertible matrices, if one prefers), $(dp|\_{H\_x})^{-1}$ is smooth and maps a smooth vector field on $M$ to a smooth horizontal vector field on $N$.
Another way of proving (b):

By problem 8-18 of Lee's Introduction to Smooth Manifolds, there's a (not necessarily horizontal or unique) vector field $Z'$ on $M$ that's $p$-related to $X$. Decomposing $Z'$ using (a) yields a smooth horizontal vector field. Uniqueness follows as stated by Yuval.