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The form $-k^2$ is just a way of saying that the sectional curvature is nonpositive. There is a nice, modern and detailed discussion of this Laplacian comparison result in P. Petersen, "Riemannian Geo …

If the (say, $d$-dimensional) base manifold $M$ is parallelizable (i.e. $TM\to M$ is trivial), then the answer to both questions is yes even globally, provided we choose a (say, torsion-free) covarian …

First of all, there is a bunch of basic things that you need to write in a slightly clearer way. If you try to topologize the set of Lorentzian metrics as you did, you need first: Restrict to the s …

This is not really an answer but rather a long-ish comment. First of all, if $M$ is not compact, $\mathfrak{A}=C(M,\mathbb{R})$ is not really a C${}^*\!$-algebra but actually only a locally C${}^*\!$- …

Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V$ the subbundle of $E=TM\oplus T^*M$ given by the graph of the musical linear isomorphism $g^\flat:TM\rightarrow T^*M$ asso …

This can be done in certain special cases besides the usual Lie derivatives along a vector field. More precisely, let $X$ and $Y$ be tensor fields over a manifold $\mathscr{M}$. The Lie derivative $\m …

The meaning of higher-order derivatives in differential geometry is better understood through jet bundles. The covariant derivative $\nabla\phi$ of (say) a smooth section $\phi:M\rightarrow E$ of a ve …

There is also the book by Daniel W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold (American Mathematical Society, 2000). Like Elworthy's book, Stroock strives to rely on g …

This is just an attempt to elaborate a bit on Ben McKay's answer beyond the confines of a mere comment. Principal $G$-bundles $P(M,G)$ over $M$ can be understood as a sort of "universal generator" of …

My answer somewhat expands on Qfwfq's above. Lie groupoids are a useful tool to reduce certain infinite-dimensional transformation groups to a finite-dimensional setting. Moreover, as put by Alan Wein …

If you consider continuous injections (resp. homeomorphisms onto their range) instead of locally Lipschitz bijections (resp. locally bi-Lipschitz), then the modified conjecture is true because of Brou …

A proof of the Poincaré lemma with optimal regularity for (non-integer order) Hölder and (nonnegative-order $L^p$, $2\leq p<\infty$) Sobolev forms is provided by Theorem 8.3, pp. 148-149 of the book b …

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\righta …

I will try to frame Will Sawin's answer in a slightly different language, that may be helpful. What you are asking for is a smooth section of $V_k(\mathbb{R}^n)$ over $G_k(\mathbb{R}^n)$, where the fo …

I do not know if it is good form for MO to cite one's own papers when answering a question, but I will take the chance. This matter is addressed in quite a bit of detail in my joint paper with Romeo B …