# 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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### Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
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### Curvature $0$ and involutive horizontal distributions

I am trying to check why curvature $0$ implies that the horizontal distribution is involutive. Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. ...
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### Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
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### Want to study Graduate Measure Theory with heavy Emphasis on Topology and/or Geometry.

I did one course in Measure Theory and want to study it again. But this time I want to do this in a way that emphasizes Measure Theoretic structure on Geometric or Topological Spaces. I don't know, if ...
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### Constant Rank Theorem for Manifolds with Boundary

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says: Formulate and prove a version of the rank theorem for a map of constant rank whose ...
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### Find sequential orthographic projections, linking three different manifolds of dimension $n=1,2,3$

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines. ...
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### Relation between de Rham Cohomology group of Lie group as a manifold and group cohomology of Lie group

Is there some relation between De Rham Cohomology group of Lie group as a manifold and group cohomology of Lie group? At first glance, they are two different things. De Rham Cohomology group is ...
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### Galois covering induces an isomorphism on the level of (co)homology

The setting is the following : we have a smooth Galois cover of manifolds $p : Y \to X$, with (Galois) automorphism group $G$. Denote by $\Omega^*(X)$ and $\Omega^*(Y)$ the spaces of differential ...
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### How weird can the boundary be so that the fundamental theorems of vector calculus hold?

Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
Given a vector field $v$ on a $2n$-dimensional manifold, how many symplectic forms are there on $M$ that make $v$ a hamiltonian vector field? Alternatively, take the set of all $(H,\omega)$ pairs, mod ...