A questions related to the Markus conjecture for special affine manifolds

An affine manifold $M$ is called special if there is a parallel volume form $\omega$ on $M$, and a nowhere vanishing vector field $\mathcal{V}.$

Here we need to point out that any affine manifold of dimension 2 will admit a nowhere vanishing vector field $\mathcal{V}.$ For the other dimensions the existence of a nowhere vanishing vector field is still open.

Throughout this section, $M$ is an affine manifold, $\nabla$ is the flat, torsionless connection in $TM,$ $\omega$ is the parallel volume form , and $\mathcal{V}$ is the nowhere vanishing vector field on $M.$ Let $M$ be a special affine manifoldfold and $p \in M.$ A positive definite bilinear form $h$ on $T_pM$ is said to be compatible with the special affine structure , if and only if, its volume form at $p$ is equal to $\omega(p)$ and $h(\mathcal{V},\mathcal{V})=1.$ The collection of all compatible billinear forms at $p$ will be denoted by $\mathcal{R}_p.$

The following lemma states that any affine manifold is locally riemannian

Lemma: Let $M$ be affine with affine connection $\nabla.$ Let $h$ be a positive billinear form on $T_pM,$ for $p \in M.$ Let $U$ be an affine coordinate neighborhood of $p.$ Then there exist a unique riemannian metric $h_U$ on $U,$ such that $$h_U(p)=h(p),$$ and $$\nabla h_U \equiv 0.$$

Taking into account the conclusion of the previous Lemma we make the following useful definition

For any affine coordinate neighborhood $U$ of $p \in M$

$\mathcal{R}_U=\left\{h_U| h \in \mathcal{R}_p \right\}$

My question is:

*If $M$ is a special affine manifold and $U$ is an affine coordinate neighborhood of $p \in M,$ then,is it true that the infimum of the injectivity radius of all metrics in $\mathcal{R}_U$ is positive?

I think I have a proof for the two dimensional case. My question is obviously motivated by the Markus conjecture for affine manifolds.

• I don't understand two points in your question: On the one hand, why is there a nowhere-vanishing vector field in dimension 2, what about $S^2$? On the other hand, it seems that the outcome of the Lemma you formulate implies a restriction on the holonomy group of $\nabla$, which is not satisfied generically. – Andreas Cap Aug 14 '15 at 11:36
• @ Andreas Thank you for the interest. the question is about affine manifolds only. the only surface that supports an affine structure is the torus. s^2 is not an affine manifold – Mike Cocos Sep 9 '15 at 22:15
• So the definition of "affine manifold" is via an atlas with chart changes that are affine transformations? If this is true, I certainly misunderstood the question. (In the terminology I am used to, this would be a flat affine manifold ...) – Andreas Cap Sep 10 '15 at 15:59
• @Andreas Sorry about the confusion. Yes what what I meant is flat affine manifold(that is a manifold that admits a flat,torsionless connection). – Mike Cocos Sep 10 '15 at 16:24