# Existence of connections in a vector bundle whose parallel transport preserves a function on a total space

Let $p:E \to M$ be a vector bundle over a smooth manifold $M$, $M\times 0$ be the image of its zero section of $p$, $\mathcal{X}(M)$ be the space of vector fields on $M$, and $\Gamma(E)$ be the space of sections of $E$.

Suppose $f:E \to\mathbb{R}$ is a smooth function such that

1) $M \times 0$ coincides with the set of critical points of $f$, however it does not necessarily coincide with the entire level set $f^{-1}(c)$ for some $c\in\mathbb{R}$.

2) the restriction of $f$ to each fibre $E_x$ over $x\in M$ has no critical points except for $x\times 0$. This in particular implies that the level sets of $f$ are transversal to the fibres.

Question: does there exist either

• a linear connection $\nabla:\mathcal{X}(M) \times\Gamma(E) \to \Gamma(E)$, or, more generally,

• a compete (=having path lifting property) Ehresmann connection $\nabla: E \to J^1(E)$, (a smooth section of the $1$-jet bundle over $E$)

on this vector bundle which preserves the function $f$ in the following sense: if $\gamma:[0,1]\to M$ is a smooth curve, $(\gamma(0), v) \in E_{\gamma(0)}$ is an any point in a fibre over $\gamma(0)$, and $\widehat{\gamma}:[0,1]\to E$ is a unique lifting of $\gamma$ with respect to $\nabla$ such that $\widehat{\gamma}=(\gamma(0), v)$, then $f \circ \widehat{\gamma}(t) = f \circ \widehat{\gamma}(0)$ for all $t\in[0,1]$.

Example when this situation appears. 1) Let $E = M \times \mathbb{R}^n \to M$ be a trivial bundle, $g:\mathbb{R}^n\to\mathbb{R}$ be a smooth function with a unique critical point at the origin, and $f:E \to \mathbb{R}$ be given by $f(x,v) = g(v)$. Then one can take a linear connection $\nabla$ to be the zero map.

2) Probably this holds if $\langle\cdot,\cdot\rangle$ is a smooth scalar product on $E$ and $f(x,v) = \langle v,v\rangle$ is the square of the corresponding norm.

At least for the tanent bundle of a Riemannian manifold $M$ the corresponding Levi-Civita connection preserves the square of the norm of tangent vectors in the above sense.

Actually I am looking for an answer in the situation when $f: N \to \mathbb{R}$ is a Bott function, $M$ a critical submanifold of $f$, and $E$ is a tubular neighborhood of $f$. Example 2) corresponds then to the case when $f$ takes minimum along $M$.

• I assume you are looking for a linear connection? Or do you just want Ehresmann connections? In the former case, just looking at the case where the fibres are one dimensional, your assumptions doesn't seem nearly enough. Jul 19, 2018 at 13:53
• Hi, thank you for reply. Yes, you are right. For linear connections the parallel transport is a linear isomophism between fibres and this is too restrictive for a general smooth function f. That should be an Ehresmann connection, i.e. just a smooth dim(M)-dimensional distribution contained in the zero of differential of f and transversal to the fibres. I repaired the question adding your remark. Jul 19, 2018 at 18:08