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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Klein Bottle embedding on $\mathbb{R}^4$ [closed]

Prove that no embedding of a Klein bottle in $\mathbb{R}^4$ can be given by a system of equations $f_1=0, f_2=0$ with independent smooth functions $f_1,f_2$ (i.e. where $df_1,df_2$ are linearly ...
7
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152 views

Green's Function for Laplacian on $S^1 \times S^2$

As indicated by the title, I am looking to find the Green's function for the Laplacian on $S^1 \times S^2$. Is such a function known? If not, does anyone have an approach to constructing such a ...
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Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\...
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How to prove this version of the fundamental theorem of calculus for curves in the closure of a domain

Dear Downvoters: if you leave a comment, you can influence the way this post gets modified, if you don't this post might never satisfy you - even though I keep editing Let $\Omega \subseteq \mathbb{...
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26 views

Principal null Direction

I need to understand what the principal null dierctions are in mathematics. Physicists define a principal null direction in a spacetime as a null vector which satisfies the Penrose-Debever equation. I ...
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1answer
44 views

Restriction of a $C^{\infty}$ vector bundle over a regular submanifold.

This question is about the content of page 134 of Tu's An Introduction to Manifolds. A $C^{\infty}$ vector bundle or rank r is a triple $(E,M,\pi)$ consisting of manifolds $E$ and $M$ and a ...
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33 views

Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
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34 views

How different definitions of connections fit together?

I'm working my way through Ivey and Landsberg, Cartan for Beginners, and I'm working on 2.6.13.2(b). After defining, for $X \in \Gamma(TM)$ a vector field, with section $s:M \rightarrow \mathcal{F}_{...
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270 views

What exactly does “differential forms are coordinate free” mean?

Most introductory texts on differential forms praise their property of allowing for a "coordinate free formulation". What exactly does this mean? What would be a concrete example for which a ...
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40 views

When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I ...
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1answer
13 views

About a proof concerning the relation of Lusternik-Schnirelman-category and cup length

In the proof of the relation between Lusternik-Schnirelman-category and Cup length (of de Rham Cohomology) for smooth manifolds from this note (theorem 2) the argument goes: Let the given manifold $M$ ...
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46 views

Zeroes of $dx_1$ on $\mathbb{R}^2$ vs. zeroes of $dx_1|_{S^1}$ on $S^1$

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ Z_{S^1}:=\{p\in\mathbb{R}^2:(dx_1|_{S^1})_p=...
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1answer
30 views

What is $u^{-1}TN$ with $u: M\rightarrow N$ be a smooth map

As picture below, $u\in C^\infty(M,N)$, $(M,g)$ and $(N,h)$ are two smooth Riemannian manifold. I don't know what mean the $\frac{\partial }{\partial y^1} \circ u$ , it is $\frac{\partial u}{\partial ...
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1answer
61 views

Laplacian via $d^*d$ in spherical coordinates over the sphere

I'm having problems to obtain the usual Laplacian on functions in Spherical coordinates over the round sphere $S^2$ using $d^*d$. My attempt: Acting on 1-forms $d^*=-*d*$ so \begin{align} d^*(\...
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1answer
55 views

Local Properties of Immersions and Submersions

This is from Bredon's Topology and Geometry, pp 83~83. He's definitions of immersion and submersion are the following: Now, in 7.3 Theorem, he explains that (using the implicit function theorem) if ...
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1answer
59 views

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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3answers
102 views

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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1answer
43 views

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 ...
3
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1answer
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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5answers
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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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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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25 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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1answer
19 views

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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0answers
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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332 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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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
28 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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1answer
39 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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0answers
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 ...
3
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1answer
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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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 ...
3
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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 $M$. ...
3
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1answer
69 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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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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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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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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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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1answer
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 ...