Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Equivalences of the definition of smooth vector fields

Let $M$ be a smooth manifold and $X\colon M \to TM$ a vector field on $M$. I'm having some trouble proving that these assertions are equivalent: (i) $X$ is smooth. (ii) for every chart $(U,\varphi) \...
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Why doesn't coordinate difference between two points correspond to distance between two points?

I know that in Euclidean geometry, where the manifold is "flat" (such that it is isomorphic to an open subset of $\mathbb{R}^{n}$), $M\cong\mathbb{R}^{n}$, one can use Cartesian coordinates, $\phi (p)\...
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Find surface in $\mathbb{R}^3$ with certain tangent spaces

By Frobenius Theorem, in $\mathbb{R}^3$ there exists a smooth surface whose tangent space is spanned by the vector fields $V(x,y,z)=(x^2+y^2,0,-y)$ and $W(x,y,z)=(0,x^2+y^2,x)$. How can I find this ...
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About de Definition of $A$-Homotopies?

Let $\pi:A\longrightarrow M$ be a Lie algebroid. An $A$-path is a path $a:I\longrightarrow A$ such that $$\sharp^A\circ a=\dot{\gamma},$$ where $\sharp^A:A\longrightarrow M$ is the anchor map and $\...
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Carolyn Gordon, David L. Webb and Scott Wolpert problem

I didn't find any reference on the subject Isospectral vs Isometry of the problem of Carolyn Gordon, David L. Webb and Scott Wolpert. Could anyone be able give me a book I may consult having a ...
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52 views

Quick question: Extension of vector bundles on a compact Riemann surface

Given the following short exact sequence of holomorphic vector bundles on a compact Riemann surface: $0\rightarrow M\rightarrow E \rightarrow N\rightarrow 0$ Fix a hermitian metric on $E$ and $n=...
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Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
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27 views

Covariant Taylor series

I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance I want to understand ...
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63 views

Misunderstanding of Atiyah-Singer

I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies Analytical index = ...
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24 views

Can terms in an integrand cancel terms in the volume element of the integral?

More specifically, given an integrand $\phi(x^{0},x^{1},x^{2},x^{3})$ of the form: $$\phi(x^{0},x^{1},x^{2},x^{3})=\frac{1}{\sqrt{\mid g_{00}\mid}}e_{0}\psi(x^{0},x^{1},x^{2},x^{3})$$ Where the ...
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Norm of the gradient of a vector field in Cartesian versus Cylindrical coordinates

It is well known that for a vector $\textbf{v}=R\textbf{e}_r+\Theta\textbf{e}_{\theta}+Z\textbf{e}_z$, its 2-norm is $\|\textbf{v}\|_2=\sqrt{R^2+Z^2}$ instead of $\sqrt{R^2+\Theta^2+Z^2}$. Now, for a ...
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26 views

Covariant derivative of parallel transport

I am learning Riemannian geometry and don't get why the following is true. We are on a Riemannian manifold with the Levi Cevita connection $\nabla$. Let $\mathcal{P}(x,x')$ be the parallel transport ...
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331 views

Intuitive interpretation of Ricci Flow

What is the best way to interpret, explain or somehow visualize the basic idea behind formal definition of Ricci Flow? I am familiar with the hackneyed expressions like "Ricci Flow is a non-...
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29 views

surface gradient of the unit normal vector

I am reading a book that defines a curvature tensor as $\boldsymbol{K} = -\nabla_\pi \boldsymbol{\hat{n}}$ where $\boldsymbol{\hat{n}}$ is the unit normal vector of a surface and \begin{align*} \...
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2answers
47 views

A question on the non trivial rank three bundle on $S^2$

We know that number of rank $3$ vector bundles on $S^2$ is just the number of equivalence classes of maps from $S^1$ to $SO(3)$. This implies that there is one and only one non trivial vector bundle ...
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1answer
27 views

Integration of differential form on ellipsoidal surface with singularity in origin

As picture below ,I want to compute the (2) , because there is a singularity in $\{0\}$ and $\omega$ is closed . So ,I have $$ \int_M\omega=\int _{\partial B_1(0)} \omega $$ I think there is a ...
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36 views

Invariant connection form

Let $\pi : E \to \mathbb{R}^3$ define a vector bundle with a connection form $\nabla$ on $\mathbb{R}^3$. The text that i'm am reading then goes on to say that $\nabla$ is $\mathbb{R}$-invariant in ...
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38 views

Determining when a differential form is closed

I'm looking at the $3$-form on $\Bbb R^4 \setminus \{0\}$ defined by $$ \gamma_k = \frac{1}{\Vert x \Vert^{2k}} i_E(dV),$$ where $k \in \Bbb R$, $E$ is the Euler vector field $x^i \frac{\partial}{\...
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25 views

Trajectories of a vector field on the 2-sphere

Consider the vector field given by given by $(-zx,zy,0)$, where we've identified $T_pS^2$ where we've identified the space of vectors orthogonal to $p$. How do we visualize the trajectories of the ...
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50 views

Perturbing a regular submanifold to ensure submersion.

Suppose $M$ is a regular compact submanifold with boundary of dimension $n$ in $\mathbb R^{n+1}\smallsetminus 0$, and assume that for each $v\in S^n$ the ray emanating from $v$ intersects $M$ in at ...
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26 views

Product of currents

De Rham currents are for differential forms as distributions are to (smooth) functions. There is a notion of exterior product for differential forms: I wonder whether there is an algebra structure on ...
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1answer
35 views

Sections of tensor bundle are tensor product of sections

Given $E,F$ vector bundles over a manifold $M$, I would like to know a proof of $\Gamma(E\otimes F) = \Gamma(E) \otimes_{C^\infty(M)} \Gamma(F)$. Where $\Gamma$ denotes the smooth sections over $M$. ...
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67 views

Vector bundles and de Rham cohomology

So $M$ is a compact manifold and I am asked to either prove the following statement or give a counterexample: if $\pi: E \rightarrow M$ is a vector bundle, then $H^2(E) \simeq H^2(M)$. I know the ...
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1answer
30 views

Differentiable functions between manifolds are continuous

Let $f:M \to N$ be differentiable function between manifolds. I want to show that $f$ is continuous. First, that $f$ is continuous should mean (correct me if I'm wrong!) that for every point $a\in N$ ...
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How to parallel transport a coframe field in a geodesic normal neighborhood?

From Chern: Lectures on Differential Geometry, page 147 Chern claimed that a torsion-free connection is completely determined locally by the curvature tensor. To show that he considered a geodesic ...
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25 views

Is this the correct way to compute tensor bundles of smooth manifolds given by a smooth function?

Let $M$ be a smooth submanifold of $\mathbb{R}^n$ given by the vanishing locus of a smooth function $f(x_1,\ldots,x_n)$. I can compute the cotangent bundle from this embedding by looking at the ...
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1answer
39 views

Construction of connections given in “From Calculus to Cohomology”

I'm struggling to understand part of the descripition of a connection on page 167 of Madsen, Tornehave "From Calculus to Cohomology". Immediately after defining a connection $\nabla$ on a bundle $\xi$ ...
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1answer
46 views

Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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15 views

Velocity of a 2-parameter curve

Let $M$ be a manifold and $I,J$ be two intervals on $\mathbb{R}$. Suppose $\alpha:I\times J\longrightarrow M$ a smooth map. It is clear that $s\mapsto\alpha(s,t_0)$ and $t\mapsto \alpha(s_0,t)$ are ...
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61 views

Maurer Cartan Form of the Heisenberg group

I'm trying to understand meaning and application of the Maurer Cartan Form, but I'm still not quite there. I'm then trying to do some examples and trying understand how it works. I begun with the ...
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1answer
29 views

The dimension of a projectivised, complexified tangent bundle

I am looking at page 15 here, starting from the line 'Let us suppose we are given a...' The Zoll projective structure is not relevant to this question, so don't worry about that. We take, in this ...
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106 views

Why are Lie groups automatically analytic manifolds?

In the book by Kolar, Michor, and Slovak, it is shown that multiplication $\mu:G\times G\to G$ is analytic in some neighborhood of $e$. Specifically, they show that in the chart given by $\exp^{-1}$, ...
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Elementary question: local computation of curvature on principal bundle

Let $G$ be a Lie group and $S=[0,1]^2$. Let $\omega$ be a connection $1$-form on the trivial principal $G-$bundle $P=S\times G$ over $S$. Let $(x_1,x_2)$ be coordinates on the base $S$. We can choose ...
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solution of under-constrained non-linear system with Implicit Function Theorem or Fixed Point Theorem

Suppose $ U \subset \mathbb{R}^n$ is open and $\mathbf{f}: U \rightarrow \mathbb{R}^m$ is $C^1$ with $ \mathbf{f}(\mathbf{a}) = \mathbf{0}$, and $\mathrm{rank}(D\mathbf{f}(\mathbf{a})) = m$. Show ...
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Relearning differential geometry

I will shortly describe my situation and than formulate the problem. From around year I am working under supervision of my professor on master thesis in differential geometry (mainly discussion of ...
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Help with derive geodesic equation

Let $(M,g)$ be a pseudo-riemannian manifold and $p,q\in M$. Suppose $\alpha:[a,b]\to M$ a smooth curve on $M$ such that $\alpha(a)=p$ and $\alpha(b)=q$. If we consider: $$L[H(s,\cdot)]=\int_a^b \sqrt{...
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1answer
41 views

Constructing symplectic structure on $T^*M$

I read the picture below, but I don't know how to get the equation above red line. Whether by using $T_{\xi_x}(T^*M)\cong T^*_xM$ ?But which isomorphism should be choice ? Then , how to check the ...
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1answer
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Reference about the surgery of Ricci flow

I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the ...
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Overview of Geometric analysis [closed]

Can anyone tell me what geometric analysis is about? After reading some articles I have a view that it uses PDE extensively for geometric problems. Am I right in this point? Also what kind of ...
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1answer
48 views

How can I equip the universal vector bundle over $\mathbf{G}(k,n)$ with a connection?

When constructing the grassmannians $\mathbf{G}_\mathbb{F}(k,n)$ for $\mathbb{F} = \{\mathbb{R},\mathbb{C} \}$ there is a natural vector bundle $$ \mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n) $$ ...
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80 views

Topology of the tangent bundle

Let $M$ be a smooth manifold, and let $\pi:TM\to M$ be its tangent bundle. We define the topology of $TM$ by declaring a subset $V$ of $TM$ to be open if and only if $\psi_\phi(V\cap\pi^{-1}(U))$ is ...
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Condition to be a Geodesic up to reparametrization

Let $M$ be an $n$-manifold and $\nabla$ a linear connection on $M$. Let $\sigma:I\subset \mathbb{R} \to M$ be curve such that in a local coordinate system we have $ \Sigma_{ij}\ (\sigma_k'' + \...
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Why use geometric algebra and not differential forms?

This is somewhat similar to Are Clifford algebras and differential forms equivalent frameworks for differential geometry?, but I want to restrict discussion to $\mathbb{R}^n$, not arbitrary manifolds. ...
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Question about Brouwer degree under uniform convergence.

I was wondering the following: Say a smooth sequence $u_k$ on a smooth manifold converges uniformly to the limit $u$. Does $u$ preserve the Brouwer degree of the $u_k$'s? I also believe this is an ...
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1answer
35 views

Geodesics on a twisted surface of revolution

How is Clairaut's Law modified to define geodesics in a twisted surface of revolution ( so not axi-symmetric) $$ u \cos v, u \sin v , f(u) + T \, v ,$$ where T is a twisting constant? It appears ...
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1answer
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Formula for the outward unit normal of a perturbed domain from D.Henry's book

I am reading D.Henry's book "Perturbation of the boundary in Boundary-Value Problems of Partial Differential Equations". At p.24, in the proof of Lemma 2.3 the following assertion is made: Let $\...
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1answer
81 views

$\hat{e_\mu } \cdot \hat{e^\nu } \neq \delta _{\mu} ^{\nu}$? Tensor algebra question.

Let $\hat{e_{\mu }}$ and $\hat{e^{\mu }}$ be the co- and contravariant basis vectors, respectively, for an arbitrary coordinate system Is it true that sometimes, $\hat{e_\mu } \cdot \hat{e^\nu } \neq \...
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21 views

Derive geodesic equation

Let $(M,g)$ be a riemannian manifold and $(U,\psi)$ local chart on $M$. If $\alpha=(x^1,\ldots,x^n)$ is a curve on $U$ such that verify: $$\sum_{i,j=1}^n \Big(\frac{1}{2}\frac{\partial g_{ij}}{\...
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22 views

Using calculus of variations to find a curve through the end points of a family of curves?

Let $I:=[0, 1]$ and let $\alpha:I\times I\longrightarrow X$ be a family of smooth curves on a smooth manifold $X$. Let us think of $\alpha$ as a family of curves $\{\alpha_s: I\longrightarrow X\}_{s\...
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1answer
58 views

Existence of closed level sets on a surface for some field

Consider an infinite 3D space with only 2 things in it: wind and a solid object. Wind evidently blows around this solid object over its rigid surface. Bascially we are trying to set up a pure field. ...