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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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Differential forms: need to understand the 1-form $dx^i$

In [1] on page 118 the authors introduce differential k-forms $\omega$ on $U \subset \mathbb{R}^n$ by \begin{equation} \omega : U \subset \mathbb{R}^n \to\Lambda^k\mathbb{R}^n \end{equation} where $\Lambda^k$ is the space of k-vectors and…
mjb
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Help with Gauss-Bonnet Theorem Application

I wish to use the Gauss-Bonnet theorem to calculate $\int_{M_r} K dA$ for the surface $$M_r = \{ (x,y,z)\in\mathbb{R}^3 |z = \cos\sqrt{x^2 + y^2}, x^2 + y^2 < r^2, x,y, > 0 \}.$$ It feels natural to change to cylindrical coordinates $(\rho, \theta,…
Peatherfed
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Show that the coordinates of $X = (d\phi)(\tilde{X})$ are $ X^i = \sum_{j}\frac{\partial u^i}{\partial \tilde{u}^j} \tilde{X}^j. $

Consider a change of variables $\phi: \tilde{U}\to U$ and let $\tilde{u}^j$ and $u^i$ be the coordinates respectively on $\tilde{U}$ and $U$. Question: Let $\tilde{X}$ be a vector field on $\tilde{U}$ with coordinates $\tilde{X}^j$. Show that the…
Alias K
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Definition of differentiable manifold.

In the notes above, I am not quite sure what the up-side-down $\Pi$ is, nor the notation of $/ \sim$. $\amalg$ means disjoint union as Peter Tamaroff advised. As for the notation of$/ \sim$, I guess it means take away subsets in the manifold that…
WishingFish
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Radial extension continuous function

I am trying to understand how a radial extension of a continuous function on $S^1$ would be. Consider $f \in S^1$ such that $f$ is continuous. It can be extended radially to a function on $\mathbb{R}^2$ such that the function is constant along the…
User987931
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A few questions regarding differential 1-forms

We define the tangent space $T_p\mathbb{R}^n$ by $T_p\mathbb{R}^n=\text{span}\{\frac{\partial}{\partial{x^1}}|_p,\dots,\frac{\partial}{\partial{x^n}}|_p\}$ and the cotangent space $(T_p\mathbb{R}^n)^*$ by…
Seth
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Definition of a circle on a manifold

I'm reading Tristan Needham's Visual Differential Geometry and Forms, specifically the start of the book where he's giving a rough intuitive idea of what non-Euclidean geometry is. He gives the example of a crookneck squash: The analogy for a…
Shirish Kulhari
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Commutativity of two vector fields. Prooving that $\frac{d}{dt}\mid_{t=0}\alpha (t)=[X,Y]_{p}$

If $X$ and $Y$ are smooth vector fields with flows $\phi^{X}$ and $\phi^{Y}$, then starting at some $p\in M$, if we flow with $X$ for time $\sqrt t$, then flow with $Y$ for time $\sqrt t$ and then flow backwards along $X$ and $Y$ for time $\sqrt t$,…
Tomas
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Preamble before Do Carmo's proof of the Cauchy-Crofton Formula

On page $42$ of Do Carmo's Differential Geometry of Curves and Surfaces he defines a rigid motion in $\mathbb{R}^2$ as a map $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $(\bar{x}, \bar{y}) \rightarrow (x,y)$ as Equation $(6)$ given by: $$x =…
IntegrateThis
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distance of cone $C=\{(x,y,z)| x^2+y^2=z^2, z\geq 0\}$

Suppose cone $C=\{(x,y,z)| x^2+y^2=z^2, z\geq 0\}$ is given and I want to compute the distance of two points $p,q \in C$. Let $d_C(p,q) := \operatorname{inf}_{\gamma} l(\gamma)$ where $l(\gamma)$ is the length of the curve $\gamma$ in $C$ from $p$…
phy_math
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The rotation index described by Do Carmo

In reading Section B of the Appendix of Chapter $1$ of Do Carmo's Differential Geometry of Curves and Surfaces, I have a few points of confusion. He writes (summarized): Let $\alpha : [0, l] \rightarrow \mathbb{R}^2$be a closed plane curve given by…
IntegrateThis
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Calculating $ d \Phi^{*} \omega$ and $ \Phi^{*} d\omega$

Let $\omega \in \Omega^2(\mathbb{R}^3)$ as follow: $\omega = x dy\land dz + y dz \land dx + z dx \land dy $. Let $\Phi: \Bbb R^3\to \Bbb R^3$ be given by $$\Phi(r, \phi, \psi) = (r\cos\phi \cos\psi, r\sin \phi \cos\psi, r\sin \psi) .$$ I can not…
Gabe
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Is correct this proposition of Warner's Foundations of differentiable Manifolds and Lie Groups?

Proposition 2.28 in Warner's book goes against my intuition. I guess that I'm missing something, but I think it is not totally correct. I paste it here: where $\mathcal{I}(\mathcal{D})=\left\{\omega \in E^{*}(M): \omega \text { annihilates }…
A. J. Pan-Collantes
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Property of parameterized by arc length curves

Problem: Suppose $\gamma:[0,L] \to \mathbb{R}^3$ is a closed and regular curve parameterized by arc length. Thus we have that $\|\gamma'(s)\|=1$ for all $s \in [0,L]$. Define $t(s)=\gamma'(s)$ for $s \in [0,L]$ where $t:[0,L] \to \mathbb{S}^2$.…
Filippo Giovagnini
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How many ways a surface can curve differently in different directions?

How many ways a surface can curve differently in different directions for a n-dimensional embedded submanifolds of $\mathbb{R}^m, m>n$? I think they can curve infinitely many ways but I am not quite certain: don't they have n-dimensional basis? I…
1LiterTears
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