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The signature convention $(−,+,\cdots,+)$ is more commonly used in General Relativity and Lorentzian geometry because of the desire among their practicioners to make a closer parallel to Riemannian ge …

There is a (related but not quite the same) construction which is valid for any Lie group $G$ and any closed (hence Lie) subgroup $H\subset G$ over any smooth base manifold $X$ which may be helpful. O …

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 …

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 …

Yes, there is. The (Constant) Rank Theorem for Banach spaces is Theorem 2.5.15 of the book of R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications (3rd. edition, Springer …

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 …

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 …

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${}^*\!$- …

Edit: Right now I will not give a complete argument in what follows (the previous version of the answer missed the part of the question asking for $S$ to be the boundary of a spacelike hypersurface in …

A good starting point for the topics you want to study, considering your stated background, is the book by Barrett O'Neill, Semi-Riemannian Geometry (Academic Press, 1983), especially Chapter 4. That …

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 …

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 …

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 …

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 …

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 …