2
votes
0answers
21 views

Generalizations of the product neighborhood theorem

As far as I know, the Product Neighborhood Theorem for smooth manifolds states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization ...
1
vote
0answers
33 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
1
vote
1answer
67 views

Reference about Gauss-Bonnet-Chern theorem.

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
2
votes
1answer
60 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
2
votes
1answer
78 views

Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
1
vote
1answer
75 views

Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
0
votes
1answer
71 views

Why $\Sigma$ is minimal, if $\frac{d}{dt} |_{t=0} \mathrm{Area}(\Sigma_t)=0$?

In this work http://arxiv.org/pdf/1204.2883v1.pdf Martin Li claimed that $\Sigma\subset M$ is minimal and $\Sigma$ meets $\partial M$ orthogonally along $\partial \Sigma$ if, only if, $$0 = ...
1
vote
1answer
76 views

Orientability of $P_{\bf R}T{\bf RP}^{2n}$

I know the following fact : (1) $ {\bf RP}^{2n}$ is non-orientable. (2) $ {\bf RP}^{2n-1}$ is orientable. (3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable. (4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
0
votes
0answers
72 views

Existence of Solution: Embedding from 2D Euclidean space to a circle

Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
3
votes
1answer
131 views

smoothness of hopf fibration projection with respect to standard differentiable structure on unit sphere

We know that steographic projection defines a differentiable structure on $S^n$ by sending points on $S^n$ to hyperplane $\{x^{n+1}\}=0$. In fact, stereographic projection $\sigma_P: S^n- P\to R^n $ ...
3
votes
2answers
580 views

Divergence theorem on Hyperbolic space

Given a vector field, say $F$, defined on a manifold $U$, the divergence theorem states that: $$\int_U\nabla \cdot F dV=\int_{\partial U} F d \Sigma .$$ Well if the manifold is $\mathbb R ^n$ and $F$ ...
5
votes
1answer
134 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
2
votes
1answer
276 views

control of the $C^{1}$ norm of a diffeomorphism

Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy: ...
1
vote
1answer
105 views

How to conclude that a path is non-trivial element of $\pi_1(M)$

Let $M^3$ be a compact manifold. If $\mathbb{RP}^2$ is embedded in $M$. Suppose, by contradiction, that $i_\sharp: \pi_1(\mathbb{RP}^2) \longrightarrow \pi_1(M)$ is non-injective and that the normal ...
2
votes
1answer
205 views

How show that a surface embedded is non-orientable?

Let be $M$ a compact $3$-manifold. If $\Sigma$ is a embedded surface in $M$, such that $\Sigma$ is homeomorphic to $\mathbb{RP}^2$. If $i: \pi_1(\Sigma) \longrightarrow \pi_1(M)$ is not injective, ...
3
votes
1answer
47 views

Possible lengths of geodecics

Let $M$ be a compact manifold (w/o boundary). Suppose that there is no closed geodesics on $M$ of length precisely $C$. I am trying to prove that there is an open cover $\{U_j\}$ of $M$ and $\epsilon ...
1
vote
2answers
293 views

Explain the Stokes -theorem from differential from into Integral form

I want to understand the Stokes -theorems deeper. I am trying to understand the operation from $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ to ...
1
vote
1answer
68 views

Variations in a Riemannian Manifold

Let be $M$ a Riemannian manifold and $X,Y$ vector fields over $M.$ Now take $p\in M$ arbitrarily, my question is, how construc a variation $f:U\to M,$ $$U\subset ...
14
votes
1answer
293 views

Question about the proof of the index theorem appearing in Milnor's Morse Theory

I am trying to get through the proof of the index theorem. The background: I have been stuck for quite a while on the following point which Milnor says is evident: Let $\gamma: [0,1]\rightarrow M$ be ...
10
votes
2answers
338 views

Reversing the Ricci flow

Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest paths on the surface. If ...
11
votes
1answer
288 views

Smooth Poincaré Conjecture

One of my professors wrote the following open question on the blackboard: If $M$ is a compact, connected smooth $4$-manifold such that $\pi_1(M) = 0$, $\pi_2(M) = 0$ (first two homotopy groups are ...