# Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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### Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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### $C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
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### Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
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### The heat kernel as a distance metric on manifolds

I recently came across Varadhan's formula (see e.g. , , , , ): $${d_{\text{g}}(x,y)^2}{} = -\lim_{t \rightarrow 0} 4 t \log K_t(x,y)$$ where $d_\text{g}$ is the geodesic distance ...
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### Relationship between Stokes's theorem and the Gauss-Bonnet theorem

Stokes's theorem and the Gauss-Bonnet theorem are clearly very spiritually similar: they both relate the integral of a quantity $A$ over a region to the integral of some quantity $B$ over the boundary ...
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### What is the essential difference between classical and quantum information geometry?

This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory. I have a ...