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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.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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Exercises to help in the understanding of differential forms?

I've been trying to grasp the concept of differential forms, which I have been encountering while studying the text "Geometric Measure Theory" by Frank Morgan. Unfortunately the explanation is very sparse and while the internet contains many…
Mike Flynn
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Using Stokes' theorem to define the exterior derivative operator

In the excellent paper "Differential forms and integration" by Terence Tao, the author has mentioned that "... one can view Stokes' theorem as a definition of the derivative operation $\omega\rightarrow d\omega$, thus differentiation is the adjoint…
Dmitry
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Smooth curve with no Frenet frame

Let $\gamma: I \rightarrow \mathbb{R}^n$ be a smooth curve. We define a Frenet frame to be an orthonormal frame $X_1, \ldots X_n$ such that for all $1 \leq k \leq n$, $\gamma^{(k)}(t)$ is contained in the linear span of $X_1(t), \ldots, X_k(t)$, $t…
user7387
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Why the tangent bundle is Hausdorff?

I was reading the lemma 4.1 in "J.M.Lee - Introduction to smooth manifolds" which says that given a smooth $n$-manifold $M$, then the tangent bundle $TM$ is a smooth $2n$-manifold. If $\pi: TM\rightarrow M$ is the natural projection, given an atlas…
Dubious
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Curvature of geodesic circles on surface with constant curvature

I am trying to solve the following exercise: Prove that on a surface of constant curvature the geodesic circles have constant curvature. "Constant curvature" in case of the surface I take to refer to the Gaussian curvature. Now, the geodesic…
koletenbert
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A question about Riemann curvature tensor and metric tensor

The Riemann curvature tensor can be expressed as: $$R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} -…
Riccardo.Alestra
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Smooth chart in what sense?

I have a question concerning smooth manifolds. As far as I've understand a smooth manifold is a pair of a manifold and a smooth atlas. Where smooth atlas means that the transition functions defiened on overlapping charts are smooth from…
harajm
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Spivak's "Differential Geometry" Volume 1, Chapter 1 ,Problem #20 part (b)

Problem 20 part (b) of Chapter 1 asks us to show that the infinite-holed torus is homeomorphic to the "infinite jail cell window." His hint helped me to get started (I think). (I apologize for not having a diagram, but one can be found by going…
user26555
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Shortest path on a sphere

I'm quite a newbie in differential geometry. Calculus is not my cup of tea ; but I find geometrical proofs really beautiful. So I'm looking for a simple - by simple I mean with almost no calculus - proof that the shortest path between two points on…
krirkrirk
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Question about orientable manifolds

Let $M$ be a connected orientable smooth manifold. Is it true that $M$ must have only 2 orientations? If yes, why?
Jr.
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A question about Killing vector and Riemann curvature tensor

In Sean Carroll's Spacetime and Geometry, a formula is given as $${\nabla _\mu }{\nabla _\sigma }{K^\rho } = {R^\rho }_{\sigma \mu \nu }{K^\nu },$$ where $K^\mu$ is a Killing vector satisfying Killing's equation ${\nabla _\mu }{K_\nu } +{\nabla _\nu…
Siyuan Ren
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Lie group action from infinitesimal action

I would like to ask, how to deduce a Lie group action, from infinitesimal action of its Lie algebra (the so called Lie-Palais theorem). More precisely, given a differential manifold $M$ and a Lie group $G$ with Lie algebra $\mathcal{G}$. Suppose we…
amine
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Dense curve on torus not an embedded submanifold

In reference to Showing a subset of the torus is dense, the responders helped show the poster that the image set $f(\mathbb{R})$ is dense in the torus. But, it's not immediately clear to me why the image set is not an embedded submanifold. If…
Winston
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Isometry Group of a Manifold

Let $(M,g)$ be a Riemannian manifold and let $I = Iso(M)$ be the group of isometries of $M$. Suppose that $I$ has no subgroups. What does this tell us about $M$?
Wintermute
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Horizontal bundle and the notion of connection

Jost explains (in Riemannian Geometry and Geometric Analysis) the term connection with the direct sum of the tangent space to a vector bundle $T_eE=V_e\oplus H_e$ as follows. Let $E$ be a vector bundle with connection $\nabla$, $e\in\Gamma(E)$ and…
gofvonx
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