6

The Main Lemma 2.2 in "Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds" (http://link.springer.com/article/10.1007/BF01233434) says essentially that there is a uniform lower bound for the harmonic radius in term of Ricci curvature bound and lower bound of injectivity radius.
The paper "E. HEBEY & M. HERZLICH, ...

5

Completeness of a pseudo-Riemannian manifold (or a manifold with affine connection, more generally) means precisely the completeness of its geodesic flow on its tangent bundle. But the tangent bundle of any covering space is a covering space of the tangent bundle. The geodesic flow is the flow of the geodesic vector field on the tangent bundle. A vector ...

3

If a compact Kähler manifold $M$ admits a holomorphic affine connection, its Atiyah class and therefore all its Chern classes are zero. By Yau's solution of the Calabi conjecture, this implies that a finite covering of $M$ is a complex torus.

3

It looks like this is from the Lecture notes on Ricci flow from Peter Topping. He mentions that he uses the symbol $T$ as the symmetric, positive definite bilinear form and also for the map $\Gamma(TM^\ast) \to \Gamma(TM^\ast)$ induced by $T$ and the metric $g$ in the following way
$$
T(\alpha)(Z)=T(\alpha^\#,Z)
$$
where $\alpha^\#$ is the dual vector field ...

2

Edit I just realised your misconception: the source and target maps are automatically surjective since they both have a section, namely the unit map. So asking that they are surjective submersions or just submersions are equivalent.
This is the only place where I saw this kind of requirement.
It's been literally the definition since the 1980s.
If the ...

1

For surfaces, see
MR1151746 Colin de Verdière, Yves. Comment rendre géodésique une triangulation d'une surface? Enseign. Math. (2) 37 (1991), no. 3-4, 201–212.

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