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Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for ...

Let $M$ be a compact Riemannian manifold with boundary, and let $E \to M$ be a Hermitian vector bundle, endowed with a compatible connection. Let $\tilde M$ be a Riemannian double of $M$. Does $E$ ...

I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...

I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is: Gromov, ...

Let $M$ be a smooth manifold and $S\subseteq M$ a properly embedded smooth submanifold. Suppose that we have a fibre metric on $TM|_S$, i.e. a positive definite real inner-product on $T_pM$ for all $p\...

It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle. What is known about the ...