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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When is a regular map a covering map?

Let $M$, $N$ be two manifolds of the same dimension. A map from $M$ to $N$ is regular provided its tangent map is one to one. A map from $M$ to $N$ is a covering map provided each point in $N$ has a ...
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74 views

Differential Geometry review questions. Need help

I have a final coming up in Differential Geometry and we got a review worksheet and I am having serious trouble with two problems. I'm still chugging along at them but I need help understanding. I ...
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3answers
48 views

Proving an identity related to the torsion of a connection.

Let $\nabla$ be a connection, and let $T(X,Y) = \nabla_{X}Y - \nabla_{Y}X - [X,Y]$ be the torsion of $\nabla$. I am trying to prove that if $f$ is a smooth function, then $fT(X,Y) = T(fX,Y)$. Using ...
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1answer
159 views

Differential geometry-geodesics

Let $\nabla_1$ and $\nabla_2$ be two affine connections of manifolds $M_1$ and $M_2$ and $\nabla$ induced connection on product $M_1\times M_2$. Prove the following: If $\gamma_1$, $\gamma_2$ are ...
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0answers
178 views

Volume form on $\mathbb{S}^2$

Let $\omega = x_1 \,dx_2 \wedge dx_3 + x_2 \,dx_3 \wedge dx_1 + x_3\, dx_1 \wedge dx_2$, with $(x_1,x_2,x_3) \in \mathbb{S}^2$, be a volume form on $\mathbb{S}^2$, and let $f : \mathbb{S}^1 \times ...
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103 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
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1answer
156 views

Where is the Volume/Surface Area of a Hypersphere?

I know that (as the dimension increases) the volume of a hypersphere concentrates near the equators and near the surface, and that the surface area (SA) concentrates near the equators as well. What ...
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1answer
75 views

parametrize curve rotating about a line

I'm thinking of parametrizing a surface of revolution created by rotating $y=x^3, 0<x<1$ about the line x = 1. My attempt is let $z=x^3$ and $|x-1|$ be the radius of circle generated by ...
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1answer
308 views

Riemann tensor symmetries

The Riemann tensor has its component expression: ...
4
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2answers
242 views

Where does the invariant expression for the exterior derivative come from?

So I've just spent about four TeXed pages (plus about a dozen TeXed pages of discarded work) proving the identity \begin{align*} d \omega(\zeta_1, \ldots, \zeta_{k+1}) &= \sum_{i=1}^{k+1} ...
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42 views

Minimal immersion

Let $N$ be an $n$-dimensional manifold immersed in an $l$-dimensional manifold $L$ by immersion $\iota: N\to L$. And let $\iota$ be minimal immersion. On the other hand, let $\phi: M\to N$ be totally ...
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95 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
2
votes
1answer
160 views

Connections and covariant derivatives

Let $A$ be a connection on a principal $G$-bundle $P$, let $\chi :G\rightarrow GL(V)$ be a representation of $G$, and let $E:=P\times _\chi V$ be the associated gauge bundle. Then, there is a ...
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1answer
86 views

Gauss-Bonnet theorem question

I was wondering if someone here can give me a hand with the proof in the image below. This is not HW, just a brain-teaser I am working on. Prove. $2\pi \chi(M)=\sum\limits_{v_i}k(v_i)$
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1answer
113 views

Stokes' Theorem: line integrals around 2-faces of n-dimensional surface?

Suppose we have a convex polytope in $n$ dimensions and are trying to calculate the surface integral (over this polytope) of some scalar function $f:R^n \rightarrow R$. Suppose all edges and vertices ...
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1answer
54 views

Maps of $G$-bundles

A vector bundle is a $GL(\mathbb{R}^m)$-bundle with fiber $\mathbb{R}^m$. A principal $G$-bundle is a $G$-bundle with fiber $G$ (where the "$G$" in "$G$-bundle" embeds into $\mathrm{Aut}\, [G]$ by ...
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119 views

Volume of “deformed torus”

I'm trying to find explicit form of volume of "deformed torus": Suppose we have a curve $\gamma(t)$ in $\mathbb{R}^n$, $t\in[0,1]$. The curve closed and smooth : ...
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1answer
52 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
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1answer
32 views

Matrix multiplication in $SO(3)$ that fixes row

I want to find all matrices $G \in SO(3)$ that do not change the first row of elements in $SO(3)$ when right multiplying by $G$, i.e. $$ \{ G \in SO(3): \forall A \in SO(3) \quad A = \begin{pmatrix} ...
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0answers
249 views

Prove that circle is the only curve which spherical indicatrix coincides with it

The task is to prove that a space curve and its spherical indicatrix of tangents coincide if and only if the curve is a circle. Def. As a point moves along a space curve C envision a unit vector t ...
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1answer
53 views

Why does these curves are different at the origin.

Let $\alpha:\mathbb R\to \mathbb R^2$ be given by $\alpha(t)= (t^3,t^2)$. The trace of $\alpha$ is drawn below: Since $\alpha'(t)=(3t^2,2t)$, we have in $t=0$: $\alpha (0)=(0,0)$ and ...
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1answer
136 views

Map of constant rank 1

A friend asked me a question which seems to be easily implied by the normal form of constant rank maps but it may be not so obvious. Let $f$ be a smooth map from $\mathbb{R}^2$ to itself, of constant ...
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40 views

FLRW metrics (isotropic and homogeneous space)

Consider a spacetime with metric $$ ds^2 = -dt^2 + a^2(t)d\Omega_k^2, \quad k=0,\pm1$$ where $a(t)$ is any regular function and $d\Omega_k^2$ is the 3-dimensional metric of the 3-sphere $S^3$, if ...
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1answer
132 views

Calculating the area and length of sets using a Riemannian metric on the sphere

Let $S^2\subseteq \mathbb{R}^3$ be the unit sphere. Let's define the Riemannian metric to be $d(x,y)=\angle(x,y)=\arccos(x,y)$. Calculate the area and circumference of the ball $B(x,R)=\left\{y\in ...
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1answer
85 views

Gradient of a function and inverse of metric

Having some knowledge in differential geometry on $\mathbb{R}^n$, I'm reading a book on Information geometry by Amari. Let $S=\{p_{\theta}\}$, $\theta=(\theta_1,\dots,\theta_n)$ be an $n$-dimensional ...
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2answers
133 views

A complete Riemannian manifold admits cutoff functions with uniformly bounded first derivatives

I'm reading a paper which uses the following fact; it appears to be standard but I am not sure where to look for a proof. Claim. Let $M$ be a complete Riemannian manifold (assumed to be second ...
6
votes
1answer
494 views

Euler characteristic is equal to self-intersection number of zero-section?

As I recall (from Guillemin and Pollack "Differential Topology") the Euler characteristic of a (for my purposes, compact and oriented) smooth manifold X is defined as $\chi(X)=I(\Delta,\Delta)$, ...
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votes
2answers
144 views

Showing that $S^1$ is orientable

I have a very silly question. While showing that $S^1$ is oriented we use two stereographic projection from the north and south pole. I have the atlas and everything. However, I just could not figure ...
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2answers
127 views

differential map of a function between smooth manifolds

Let $f:M \rightarrow N$ be a function between manifolds. If $df_p=0$ at all points $p \in M$, can we say that $f$ is a constant function? I think it would be but have not been able to prove it since I ...
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1answer
48 views

How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
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0answers
52 views

Smooth parametrizations of continuous curves

Consider the curve $t\to (t,|t|)$ in $\mathbb R^2$. Even if it has a cusp in $(0,0)$ i can reparametrize it with a smooth function. Take for example $$ t\mapsto \begin{cases}(\mathrm e^{-1/t},\mathrm ...
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0answers
55 views

Heisenberg manifold

I am interested in the Heisenberg manifold, which is the quotient of the real Heisenberg group by the discrete Heisenberg (sub)group. It is a 3 -manifold which may be viewed as a circle bundle over ...
2
votes
1answer
70 views

Family of diffeomorphisms on a Manifold

I am reading Huybrechts book Complex Geometry. On page 259, below Def.6.1.3: the author talks about families of diffeomorphism and then he gives a power series expansion and uses it to define a ...
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58 views

Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let ...
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0answers
67 views

Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
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1answer
105 views

Line bundles over $\mathbb R P^2$

As in this post, I'm continuing studying line bundles. Now it's line bundle over $\mathbb R P^2$. I know that this bundle is not trivial. So I want list up to equivalence all bundles over $\mathbb R ...
3
votes
2answers
128 views

Line bundle over $S^2$

I'm trying to study line bundle over $S^2$. In this post was outlined the method based on clutching functions. But now I'm interesting in another approach. For the sphere there is two maps : upper ...
2
votes
2answers
66 views

Derivative: a special tangent

I've learned in Euclidean Geometry that the tangent is a line which pass through only a point. For example, if someone ask me to find the tangent at this point $A$, I can easily say that the tangent ...
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vote
1answer
202 views

Higher order differential on a manifold, connections

I am trying to understand how to define the second order differential of a map $f : M \rightarrow N$ between smooth manifolds. I came across this exact same question : Higher-order derivatives in ...
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0answers
82 views

Homotopy versus path-homotopy on punctured surface

I have some problems with homotopies. The situation is this: Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open ...
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1answer
46 views

An isometrie $\varphi: S_1\to S_2$ which cannot be extended into distance-preserving map

I'm searching for an example isometrie $\varphi: S_1\to S_2$($S_i$ are regular surfaces) which cannot be extended into distance-preserving maps $F: \Bbb R^3 \to \Bbb R^3$. A reference or hint will ...
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vote
1answer
68 views

Why is this combination of a covariant derivative and vector field a (1,1)-tensor?

I have a question regarding something Penrose says in section 14.3 of The Road to Reality. It says '...when $\nabla$ acts on a vector field $\xi$, the resulting quantity $\nabla \xi$ is a ...
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1answer
96 views

Does local flow of left-invariant vector field commute with the left-translation operator?

Let $G$ be a Lie group and $X$ a left-invariant vector field over $G$ (i.e. $\forall g,p\in G: (D_p l_g)(X_p) = X_{gp}$ whereby $l_g$ is the map $G\rightarrow G:p\mapsto gp$). Let $\phi_t$ be the ...
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1answer
30 views

Reference on diffeomorphisms between subspaces of ${\mathbb R}^n$

I am looking for a basic reference on Diffeomorphisms. I am mainly interested on checking conditions to tell whether or not given two spaces $M,N \subseteq {\mathbb R}^n$, there exists at least one ...
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2answers
112 views

Vector field invariant under transitive action: restricts to free transitive action?

Thinking about how you can put vector fields on homogeneous spaces that respect the homogeneity, I'm interested in the following situation: Let $V$ be a nonzero vector field on a manifold $M$, let ...
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votes
3answers
658 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
2
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1answer
46 views

Compact hypersurface of $\mathbb{R}^{n+1}$ with positive curvature is diffeomorphic to $S^n$.

I have a compact hypersurface $M$ of $\mathbb{R}^{n+1}$ with positive curvature. I need to show that it is diffeomorphic to $S^n$. The hint is to consider the shape operator $A_{\nu_p} x$, where ...
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3answers
145 views

Smooth map from $S^1$ to $S^1$ with degree zero

Is there any example of a smooth map $f:S^1\to S^1$ that has degree zero that is not the constant map? Either the map would have no regular values or every regular value would have an even number of ...
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1answer
144 views

A specific example of $F$-related vector fields

I need to prove the following: Let $F:\Bbb{R}\to\Bbb{R}^2$ be the smooth map $F(t)=(\cos t,\sin t)$. Then $d/dt\in\mathcal{T}(\Bbb{R})$ is $F$-related to the vector field $Z\in\mathcal{T}(\Bbb{R}^2)$ ...
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1answer
28 views

Prove that an immersion is minimal

If we have an immersion from $R^{2}\to R^{4}$, given by $(x,y)\to (x,y,x,y)$, how can we prove that the given immersion is minimal or not. Suppose that we choose $\{e_{1}, e_{2}\}$ for basis of ...