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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107 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
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1answer
106 views

Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
7
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2answers
268 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
2
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1answer
349 views

What is a tangential gradient?

If $\mathbb{R}_{+}^{n}$ is the half space $\{x\in\mathbb{R}^n\vert\ x_n>0\}$ and $u$ is a twice differentiable function in $\mathbb{R}^{n}$. If we write: \begin{equation} ...
7
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1answer
292 views

How can a $C^1$-continuous surface have infinite curvature?

Short version: Apparently it is possible for a $C^1$-continuous parametric surface to (locally) have infinite (Gaussian) curvature. I find this quite counter-intuitive, because I always thought that ...
2
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1answer
98 views

Surfaces without conjugate points

I'm trying to understand some aspects of the geodesics of surfaces without conjugate points and the following question came out: consider the hyperbolic space and two geodesics wich are asymptotic to ...
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0answers
96 views

How to construct an orthogonal coordinate system from a smooth planar curve?

Given a planar curve $$\gamma:\mathbb R\to\mathbb R^2, t\mapsto \gamma(t) \text{, normalized to } |\gamma'(t)|\equiv1,$$ the tangential vector $$T(t) = \gamma'(t)$$ and the normal vector $$N(t) = ...
2
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1answer
109 views

If $ \eta $ and $ \varphi $ are closed differential forms, then prove that $ \varphi \wedge \eta $ is a closed differential form.

Let’s assume that $ \eta $ and $ \varphi $ are closed differential forms. Then how can I prove that $ \varphi \wedge \eta $ is a closed differential form as well? Please explain how to solve this ...
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3answers
365 views

How to differentiate a differential form?

Please explain me the idea of differentiating differential forms (tensors). Example: compute d(xdy + ydx) The answer is known, we should have 0. What's the rule?
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1answer
117 views

linear equivalence of divisors given by sections of a linebundle

The topic of divisors is new to me and I wonder if I understand a few things correctly. In my situation I have an analytical Manifold $M$ of dimension $2$ and its compactification $\overline M$. I ...
4
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1answer
280 views

Do Carmo: Linear Killing field anti-symmetric?

In Exercise 3.5a of Riemannian Geometry, do Carmo defines a vector field $v$ on $\mathbb{R}^n$ to be linear if it's linear as a map $v\colon \mathbb{R}^n \to\mathbb{R}^n$. He then asks the reader to ...
9
votes
1answer
130 views

Characterizing singularities using sheaves of smooth functions

Short version: Let $H\subset M$ be a closed subset of a smooth manifold. Equip $H$ with the sheaf $\mathcal{F}$ of smooth functions (so that a section over an open $U$ is the restriction to $U$ of ...
2
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1answer
286 views

Solve the Poisson Equation on a Riemannian Manifold

Imagine that I have a field that obeys to the Poisson equation. To solve the equation, in my bag of tools I only have the divergence theorem or the Fourier/Laplace transform. They usually are enough ...
6
votes
1answer
201 views

Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
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0answers
88 views

Why does the map $x^2$ have constant rank?

I'm just trying to wrap my head around the rank of a map via some examples. Now, if I have the smooth map of manifolds $F:\mathbb{R} \to \mathbb{R}, F(p) = p^2$, then the differential is given by ...
4
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2answers
265 views

Property of normal coordinates

Let $M$ be a Riemannian manifold and $\nabla$ the Levi-Civita conection. I need to prove the following. Let $B$ be an open ball of radius $r$ in $T_pM$ such that $\left.exp_p\right|_B$ be a ...
2
votes
1answer
94 views

Is $\zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3}$ exact in the complement of every line through the origin?

$r=\sqrt{x^2+y^2+z^2}$ of course. If the line is the $z$ axis, it is given in the book (Rudin) that $\zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right)$ I've managed to figure out 2 ...
5
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1answer
196 views

Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
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0answers
188 views

Difference between distance between two points and metric

if i have a line element given e.g. $ds^2=\frac{dx^2+dy^2}{2y} $ is it then always possible to derive a distance between two points in this metric? and how would one determine the length of a curve if ...
0
votes
1answer
54 views

Showing a Frame is Orthonormal

I know this is really simple. But I don't know how to carry out the calculation, I can only "see" why the following is a orthonormal frame. Let $$E_1 = \frac{x}{r}\frac{\partial }{\partial x} + ...
0
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1answer
65 views

Convert line integral between different metrics?

If I have $$ \int\limits_0^T \frac{\sqrt{\dot{x}(t)^2+\dot{y}(t)^2}}{\sqrt{2 y(t)}}dt $$ I can convert this problem of finding the solution to the brachistochrone problem to a geometric problem by ...
3
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2answers
2k views

Left-Invariant Vector Field of a Lie Group

How do I tell if a vector field on a Lie Group is left-invariant? I have the technical definition. But, I want to understand given a specific vector field what should I do to test if it is ...
1
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1answer
143 views

Smoothness Criterion for Vector Fields

I'm going to just write the proof (straight from Lee), my question is about the $(*)$ stared part. Let $M$ be a smooth manifold and let $X:M \to TM$ be a (rough) vector field. If $(U,(x^i))$ is any ...
2
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2answers
142 views

what is $C^{-\infty}(\mathbb{R})$

Thanks in advance. what is $C^{-\infty}(\mathbb{R})$? Is that the same as the "distribution" defined in differential geometry? It would be helpful if someone can describe it in another way ...
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1answer
38 views

Closure of a set's cone

working in $\mathbb{R}^3$ , say I'm looking at the set : $\{1\} \times S^1$ denote E for the cone of the set above, is E a closed set? (I think it is) if not, what is it's closure? thanks.
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0answers
34 views

How is integration of differential form defined as, and how to calculate it

How is integration of differential form defined as? And how does one calculate the value of integration?
4
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1answer
224 views

The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)

In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it. Let $\gamma: ...
7
votes
2answers
679 views

How to embed Klein Bottle into $R^4$

I am using Do Carmo's Riemannian Geometry, and struggling to solve a problem. The problem is: Show that the mapping $G:\mathbb{R}^2\to\mathbb{R}^4$ given by $$G(x,y)=((r\cos y+a)\cos x,(r\cos ...
1
vote
2answers
71 views

Pullback map and its equality consequence

On my textbook, it says: $$F^{\star}(dy^i) = \sum_{j=1}^{n} \frac{\partial y_i}{dx_j}dx_j$$ where $F^{\star}$ is a pullback map, map $F: M_1 \rightarrow M_2$ and $dx_j$s are forms on $M_1$ and ...
5
votes
2answers
69 views

Tangent Space of $\operatorname{Aut}(T_e G)$

Let $G$ be a Lie Group, how to prove that the tangent Space of $\operatorname{Aut}(T_e G)$ at identity is $\operatorname{End}(T_e G)$? Thanks.
1
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1answer
100 views

Vector space structure on $(-1,1) \subset \mathbb{R}$ (or: möbius strip as vector bundle)

I'm first putting the question into it's context, so probably you can see if i'm asking the wrong question to get what i want. The Task is to show that the Möbius (Moebius) strip is a Vector bundle ...
7
votes
1answer
168 views

Set theoretic implications of constructions in Differential Geometry/ Topology

In subjects like Differential Geometry/ General Topology one often constructs for each $x$ in a space $X$ a set $U_x$ satisfying certain properties. Examples where one does constructions like this: ...
0
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1answer
64 views

smooth connection on exterior powers of vector bundles

Suppose E is a smooth vector bundle with a smooth connection ∇. Then it induces a smooth connection on tensor powers of E. Does it also induce smooth connection on say exterior powers of E. thanks.
1
vote
1answer
157 views

Show that the vector field $\operatorname{grad}f$ is smooth

Let $M$ be a Riemannian manifold and $ f:M\rightarrow\mathbb{R}$ be a smooth function. Define a vector field $\operatorname{grad}f$ in $M$ as $$\langle\operatorname{grad}f,\,V\rangle=df(V)$$ for all ...
7
votes
1answer
396 views

Vector field and integral curve

Let M a riemaniann manifold, $V$ a vector field in M, and $\phi_t$ the respective flow Let $p\in M$, and $\gamma$ the integral curve of $V$. Show that for all $v\in T_pM$ ...
3
votes
1answer
509 views

Image of Homomorphism of Lie groups

This is exercise from Lee: Introduction to smooth manifolds. Suppose $f \colon G \to H$ is homomorphism of Lie groups (real, finite-dimensional). Q: Is image $Im(f) \subseteq H$ a Lie subgroup of H? ...
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1answer
59 views

Reference of spin structure

I am looking for some elementary books (may be introduction) about Spin structure in general, and Spin structure on Riemannian manifolds. Someone can help me? Thanks a lot!
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1answer
45 views

Question about self homeomorphism of $\mathbb C\mathbb P^2$

Can anyone give me any idea about how to show that: any self homeomorphism of $\mathbb C\mathbb P^2$ is orientation preserving? Thanks.
21
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2answers
2k views

What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
0
votes
1answer
77 views

Non-degenerate smooth functions on a manifold

I am trying to prepare a presentation on "the use of differential geometry in the theory of critical points", but only the case where there is a single variable. (only in dimension 1), and i ask ...
10
votes
2answers
307 views

No Smooth Onto Map from Circle to Torus

My professor was lecturing today and he made this statement which I was unable to verify. (I worded it nicer) There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$. When ...
6
votes
1answer
224 views

Finding an algebra of smooth functions on a manifold with a given product.

I am having trouble with the following exercise from Jet Nestruev's Smooth Manifolds and Observables. It is one of the first exercises in the book and I have no idea how to approach this type of ...
3
votes
0answers
77 views

What is the Induced Representation in Geometric Terms

As is well known, for $G$ a Lie group, and $H$ a subgroup of $G$ such that $G/H$ is homogeneous space (or maybe this is always a homogeneous space?), we have a correspondence between representations ...
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1answer
277 views

shortest distance between two points on $S^2$

Length of Curve in $2D$ is $l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2}$ Length of a curve in $3D$ is ...
4
votes
2answers
816 views

Length of curve in 3D spherical coordinate

let $r$ be the magnitude of a vector in 3D with Spherical co-ordinate $(r,\theta,\phi)$ and cartesian coordinates is $(x,y,z)$, whose angle with $z$ axis is $\phi$ and projection of the vector makes ...
2
votes
1answer
268 views

Selecting Differential Geometry Exercises

I'm self-studying differential geometry with Do Carmo's books "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry" and I find those books very good, however I feel a little ...
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2answers
680 views

How can I calculate the Euclidian displacement of two places on a sphere (earth in this case ) and calculate the

I would like to get the formula on how to calculate the distance between two geographical co-ordinates on earth and heading angle relative to True North. Say from New York to New Dehli , I draw a ...
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1answer
147 views

Length of a curve on $S^2$

$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$? $2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates? ...
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0answers
60 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
3
votes
1answer
150 views

Show that we can define a connection on any manifold using partitions of unity

Suppose that $(U,\varphi)$ is a chart on manifold $M$, and $X,V$ are vector fields on manifold $M$, then we can write: $$X=\sum_{i=1}^{i=n}X^{i}\frac{\partial}{\partial x^{i}}$$ on $U$, and define a ...