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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 …

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 …

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 …

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 …

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 …

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 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 …

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 …

This can done by showing that the restriction of the Whitney topology to $\mathcal{C}^\infty_c(M)$ coincides with the finest linear topology that makes the inclusions $\mathcal{C}^\infty_c(K)\hookrigh …

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 …