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The Main Lemma 2.2 in "Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds" ( 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, ...


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 ...


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.


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 ...


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 ...


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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