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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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Geometric significance of the differential of the Gauss map, $dN_p: T_p(S) \rightarrow T_p(S)$, being a self adjoint linear map.

i'm studying differential geometry for the first time and I just started reading about the gauss map and I've gotta say this stuff is pretty cool. The claim made in the title of this post is from page 142 of Do Carmo's Differential geometry book,…
Math is hard
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Lie Bracket and flows

Can anyone show me how do I differentiate this? Suppose I have $\Phi^{X}_{t}$ and $\Phi^{Y}_{t}$ both flows with $X$ and $Y$ respectively starting from point $p$, what is $\frac{d}{dt}|_{t=0}\phi^{Y}_{-\sqrt{t}}\circ \phi^{X}_{-\sqrt{t}}\circ…
enoughsaid05
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Why do metrics act on tangent vectors?

Consider a manifold $M$ and a curve $\gamma : \mathbb{R} \supseteq I \rightarrow M$. The length of this curve is defined as $$ L = \int \sqrt{g(X,X)}_{\gamma(t)} dt $$ where $g$ is the metric tensor, $X$ is the tangent vector to the curve and $t$…
Hermitian_hermit
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It's confusing to calculate Euler characteristic of this surface

This pic below is an exploded view of a cone. I'm trying to calculate the Euler characteristic of the surface made from the fragment $M$, i.e., At first I thought the Euler characteristic is 0, but the one who made this question says it is…
민광건
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Explicit formula for space curves

I've been looking a bit into differential geometry and have gotten stuck on a question: Given a function $f,$ is there a way to find the explicit space curve which has $f$ as both it's curvature and torsion? I've been able to find a formula for a…
Steve
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Differential of the exponential map on the sphere

I have a problem understanding how to compute the differential of the exponential map. Concretely I'm struggling with the following concrete case: Let $M$ be the unit sphere and $p=(0,0,1)$ the north pole. Then let $\exp_p : T_pM \cong \mathbb{R}^2…
alexlo
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Let $X_M^c$ be the space of vector fields on a manifold $M$ with compact support. Prove that $X_M^c=[X_M^c,X_M^c]$

Let $X_M^c$ be the space of vector fields on a manifold $M$ with compact support. Prove that $X_M^c=[X_M^c,X_M^c]$. Proving that $[X_M^c,X_M^c]\subset X_M^c$ is relatively simple. However, I have not been able to prove $X_M^c\subset…
user67803
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Measuring curvature in Flatland

Gauss' Theorema egregium says that the Gaussian curvature of a surface can be determined entirely by measuring angles, distances and their rates on the surface itself. A surface looks like $\mathbb{R}^2$ locally so the angle sum of…
Hans-Peter Stricker
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Homotheties and Geodesics

A homothety is a smooth map between Riemannian manifolds $f:(M,g)\rightarrow (N,h)$ such that $f^*h=cg$ for some $c\ne 0$. My questions is: how are the geodesics on these two spaces related? Can we say that the geodesics on $(N,h)$ are given by…
Oliver Jones
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Distinguished map that straightens a curve

Consider a smooth finite curve $\gamma$ without intersections in $\mathbb{R}^2$. Consider the family of smooth maps $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that $T(\gamma)$ is a straight line. Which ways are there to distinguish one of these…
Hans-Peter Stricker
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Exterior derivative well defined on vector fields

Let $M$ be a manifold and $d$ the exterior derivative on forms. We extend $d$ to vector fields as follows: if $X = \sum_i X_i \partial_i$ in local coordinates, then $$dX = \sum_i (dX_i) \partial_i$$ i.e. if $Y$ is another vector field then $dX(Y)$…
user15464
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Approximations of shortest curves

Consider a set of functions $f:[0,1]\rightarrow \mathbb{R}$ with $f(0) = f(1) = 0$ that are supposed to approximate the function $\mathbf{0}:[0,1]\rightarrow \mathbb{R}$ with $\mathbf{0}(x) \equiv 0$. The graph of $\mathbf{0}(x)$ is to be seen as…
Hans-Peter Stricker
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Manifold of fixed points

Let $M$ be a smooth manifold and let $G$ be a Lie group smoothly acting on $M$. Then, under suitable assumptions (if $G$ acts freely and properly on $M$) we have a new smooth manifold $M/G$ corresponding to the orbits of the action. I would like to…
Benjamin
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Pullback connection and curve

My teacher defined the pullback bundle and gave me an example: Let $\gamma:(-1,1) \to \mathbb{R}^2$ be a smooth curve. Then for $w \in \Gamma(\gamma^*T_p{\mathbb{R}^2})$, the pullback connection is $$\nabla_{\frac{\partial}{\partial x}}w =…
maximumtag
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Tangent bundle of a manifold as more than a vector bundle

Let $M$ be a smooth manifold. The tangent bundle is naturally a smooth vector bundle, but it obviously has more structure than that. Specifically, there is a natural action of the the diffeomorphism group of $M$ on $TM$. Unless I am mistaken,…
Rbega
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