(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

learn more… | top users | synonyms

0
votes
1answer
712 views

Vector perpendicular to timelike vector must be spacelike?

Given $\mathbb{R}^4$, we define the Minkowski inner product on it by $$ \langle v,w \rangle = -v_1w_1 + v_2w_2 + v_3w_3 + v_4w_4$$ We say a vector is spacelike if $ \langle v,v\rangle >0 $, and it ...
4
votes
1answer
89 views

Does the $O(n)$ bundle of a manifold depend on the metric?

Let $g_1$ and $g_2$ be two Riemannian metrics on a manifold $M$. These induce two $O(n)$ bundles on $M$, whose fibers over each point $x\in M$ are the groups of orthogonal transformations of $T_x M$ ...
0
votes
0answers
72 views

Codimension one foliation

let $f:M \rightarrow \mathbb{R}$ be a nowhere vanishing function defined on a Riemannian manifold $(M,g)$. consider the distribution $\ker(df)$. This is clearly involutive and thus defines a ...
0
votes
0answers
44 views

Detail in polar action

I am reading a paper "Tits geometry and positive curvature - Fang, Grove, and Thorbergsson" See the following site http://arxiv.org/pdf/1205.6222.pdf In page 7, the 9-th line from the bottom ...
4
votes
0answers
121 views

Transforming the Dirac Operator on $S^1$

My goal is to understand as much as I can about the Dirac operator on $S^1$ where we give $S^1$ the spin structure given by the connected double cover of the frame bundle. The spinor bundle on $S^1$ ...
2
votes
1answer
193 views

Derivative of a metric tensor along a curve

Let $M$ be a Riemannian manifold with metric tensor $g$ and Levi-Civita connection $\nabla$. Also, let $u: \mathbb{R}\to T_pM$ be a smooth curve in $T_pM$. In a proof, my course notes assure that ...
0
votes
0answers
132 views

Geodesics (2): Is the real projective plane intended to make shortest paths unique?

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The ...
0
votes
0answers
91 views

Geodesics (1): Spaces with more than two geodesics between two points

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The ...
6
votes
1answer
292 views

Uniformization Theorem for compact surface

Why in proof of proposition 6 of http://arxiv.org/abs/0909.1665, they claim that if a embedded surfaces $\Sigma^2 \subset (M^3,g)$ is homeomorphic to $\mathbb{RP}^2$, where $M$ is compact manifold, ...
5
votes
1answer
167 views

Isometries from Diffeomorphisms

Given a compact, connected Lie group G of diffeomorphisms on a manifold M, how to construct a Riemannian metric on M such that elements of G are isometries of M?
1
vote
0answers
34 views

Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
1
vote
1answer
91 views

The measure of a special set in Riemannian manifold

Let $M$ be a complete Riemannian manifold and $a$, $b$ two different points on it. We define a set $A =\{x\in M | \ d(x,a)=d(x,b)\}$ where $d$ is the distance induced by the metric of $M$. My question ...
6
votes
1answer
289 views

Dirac Operators on $S^1$

I am trying to understand the Dirac operators associated to the 2 spinor bundles on $S^1.$ I have been getting very confused about why one bundle has nontrivial harmonic spinors and the other ...
2
votes
1answer
138 views

Quasiconformal map between the complex plane and a disk

According to the Poincaré-Koebe theorem, it is known that the unit disk $\mathbb D$ and the complex plane $\mathbb C$ aren't conformally equivalent. My question is maybe naive, but I was wondering if ...
3
votes
0answers
76 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
3
votes
2answers
121 views

SO(5)-invariant metrics on the 4-sphere

Are there any examples of Riemannian metrics on $S^{4} \subset \mathbb{R}^{5}$ that are not SO(5)-invariant? Or are all metrics on the 4-sphere SO(5)-invariant? Hope my question is not too trivial ...
0
votes
1answer
213 views

How can i find this Basis in $\mathbb{R}^n$?

Define a Pseudo-Riemannian Metric $g$ in $\mathbb{R}^{n+1}$ by $g(u,v)=-u_0v_0+u_1v_1+...+u_nv_n$, where $u=(u_0,u_1,...u_n)$. Let $\eta\in\mathbb{R}^{n+1}$ be a vector such that $g(\eta,\eta)=-1$. Is ...
0
votes
1answer
23 views

Is this System Solvable?

Suppose we are given a Riemannian Metric $g$ on $\mathbb{R}^n$. Let $v_1\in\mathbb{R}^n$ with $g(v_1,v_1)=1$. Is it possible to find a base $\{v_1,v_2,...,v_n\}$ of $\mathbb{R}^n$ in such a way that ...
3
votes
0answers
98 views

geodesic submanifolds

I have a question to find all geodesic submanifolds of the hyperbolic space in n-dim. I did an exercise that all geodesics must be either lines perpendicular to the boundary of the hyperbolic space ...
1
vote
2answers
116 views

How can i prove that the Hyperbolic space is complete by using Divergent Curves?

Let $$H_+^2=\{(x,y)\in\mathbb{R}^2:\ y>0\}$$ and consider the Lobatchevski metric on $H_+^2$: $$g_{11}=g_{22}=\frac{1}{y^2},\ g_{12}=0$$ How can one prove the completeness property of this space ...
3
votes
1answer
201 views

Levi-Civita connection

Well if $\Sigma$ is a submanifold of $R^{n+p}$ and $\{e_i,e_\alpha\}$ is orthonormal frame over $\Sigma$ where the $e_i$'s are tangent and the $e_\alpha$'s are normal to $\Sigma$. Can anyone prove ...
4
votes
3answers
333 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kähler condition implies that $\omega$ is closed with respect to $d$, so ...
2
votes
1answer
378 views

Second fundamental form proportional to the Hessian

Let $(M^n,g)$ be a Riemannian manifold and $f:M\to\mathbb{R}$ a smooth function. Then the graph $S=\{(p,f(p))\mid p\in M\}$ is a submanifold of $(M\times\mathbb{R},g+g_{\mathbb{R}})$ and carries the ...
1
vote
0answers
59 views

How can i show this Equality?

Let $M$ be a complete Riemannian manifold and $N\subset M$ a closed submanifold. If codimension of $N$ is $0$ take $q\in\partial N$ and $v\in T_qN$, where $\partial N$ is the boundary of $N$ as a ...
4
votes
0answers
126 views

Curvature of particular Riemannian metric

Let $U = \{ (x_1, \dots, x_n) \mid x_j > 0 \text{ for all } j\}$ and let $\|x\|^2 = \sum_j x_j^2$. The function $x \mapsto -\log \|x\|^2$ is strictly convex on $U$ and thus defines a Riemannian ...
2
votes
1answer
251 views

version of Bianchi identity

Let $F \to E \to M$ be a smooth fiber bundle with connection $\omega$ and curvature $R$. We can form a (graded) vector bundle by taking the complex of differential forms at each fiber. Call this ...
3
votes
2answers
371 views

Geodesic distance from point to manifold

This is question 1 of chapter 9 from Manfredo do Carmo's Riemmanian Geometry. $M$ is a complete Riemmanian manifold and $N\subset M$ a closed submanifold. $p_0\in M$ and $p_0\notin N$. Let $d(p_0,N)$ ...
2
votes
2answers
268 views

Tangent spaces at different points and the concept of connection

If $M$ is a smooth manifold and $TM$ is the tangent bundle clearly $T_pM\cong T_qM$ (as vector spaces) for every $p,q\in M$. Nobody ensures that the previous vector spaces isomorphism is natural (or ...
2
votes
0answers
58 views

About Thom theorem (representation submanifold for $H_{n-2}(M)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold. And in the Harper and ...
7
votes
2answers
357 views

About Gauss-Bonnet Theorem

The Gauss–Bonnet theorem say that: If $\Sigma \subset M =\mathbb{R}^3$ is a compact 2-dimensional Riemannian manifold without boundary, then $$ \int_{\Sigma} K = 2\pi\chi_{\Sigma}$$ where $K$ is the ...
1
vote
0answers
73 views

Is this mean curvature?

Suppose $N_t:=\partial B(p, t)\subset M^{n+1}$ be the distance sphere in a Riemannian manifold. Let $\{x_1, \cdots, x_n\}$ be a coordinate of the distance sphere $\partial B(p, t)$. Hence $\{x_1, ...
3
votes
0answers
151 views

Why does the Laplace operator extend to $L^2(X)$?

Suppose $X$ is a Riemannian manifold. Then we get a Laplace operator on $C^\infty(X)$. In most texts I see the Laplace operator extended to $L^2(X)$, but I don't see how, since it does not seem to be ...
10
votes
2answers
1k views

Is the Nash Embedding Theorem a special case of the Whitney Embedding Theorem?

The Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space. The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean ...
0
votes
1answer
57 views

Is this Laplacian comparison?

Cheeger-Colding-Minicozzi: 1995 Linear growth harmonic functions on complete manifolds with nonnegative ricci curvature in GAFA Page 952: From Laplacian comparison, we have for $r<R$, ...
3
votes
1answer
258 views

Relationship beween Ricci curvature and sectional curvature

Let $(M,g)$ be a Riemannian manifold and assume that for all orthonormal $v,z$ the sectional curvatures is bounded from below i.e. $K(v,z) \geq C$, where $C > 0$. Is it in this case possible for ...
2
votes
2answers
42 views

Inequality about convex function

This question is arised when I study the content in the following question entitled by " lower bound of a special type of convex functions " in here. Let $f: {\bf R}^n \rightarrow {\bf R}$ be a ...
4
votes
1answer
197 views

Why does it appear that Willmore energy is always zero?

The answer is "because I'm being sloppy," but the problem is I don't know exactly where I'm being sloppy. Here's my sloppy argument: Let $M$ be a smooth compact surface without boundary in ...
8
votes
2answers
404 views

sign error proving product rule for the Laplacian on a product of Riemannian manifolds

Given two Riemannian manifolds $M$ and $N$, of dimension $m$ and $n$ respectively, the product manifold $M\times N$ has a Riemannian structure, and there is a Laplacian operator $\Delta_{M\times N}$ ...
1
vote
1answer
146 views

Why is a Riemannian submersion a submetry

I have encountered a conclusion in the book but I do not know how to prove it. If $F$ is a Riemannian submersion from $(M,g)$ to $(N,\tilde g)$, then it is a submetry. (A mapping $F$ is a submetry if ...
2
votes
0answers
47 views

Complex Laplacian on 0-forms

Let $M$ be a complex manifold and $\Delta^{\bar \partial} = \bar\partial^* \bar\partial + \bar\partial\bar\partial^*$ the complex laplacian. Is it true that $\Delta^{\bar\partial} f = \Delta f$ (the ...
1
vote
1answer
146 views

Continuity of the orthogonal projection into tangent space.

Let $\mathcal M \subset \mathbb R^d$ be a smooth manifold, and for each $s \in \mathcal M$ let $T_s[\mathcal M]$ denote the tangent space of $\mathcal M$ at $s$. For each $s \in \mathcal M$ let $P_s$ ...
0
votes
1answer
292 views

isometry and exponential map

I got stuck on the following questions. Can anyone give me idea how to proceed? Suppose $M$ is a Riemannian manifold and $\phi: M \to M$ an isometry map. If $\phi(p)=p$ and $\phi(q)=q$ prove that ...
1
vote
1answer
122 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
0answers
57 views

Complex structure on the product of two complex Kaehler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kaehler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
2
votes
1answer
229 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, ...
2
votes
1answer
49 views

systole of space projective 3-dimensional

I'm study the papper "H. Bray, S. Brendle, M. Eichmair, and A. Neves, Area-minimizing projective planes in three-manifolds, Comm. Pure Appl. Math". (see http://arxiv.org/abs/0909.1665). Let be ...
0
votes
1answer
200 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
1
vote
0answers
25 views

Special geodesic unit speed curves on a Riemannian manifold subtending one another

Let $f(s,t)$ be a two times continuously differentiable piece of a surface, which is part of a Riemannian manifold $M$. Let $0\leq s\leq 1$ and $-\epsilon<t<\epsilon$ be given such that all ...
3
votes
1answer
91 views

Geodesics in a manifold M diffeomorphic to $\mathbb S^2$

I am now reading the book Calculus of Variations written by Jost and I encountered the following problem (in Theorem 2.3.3.): Let $M$ be a differentiable submanifold of $\mathbb R^d$ diffeomorphic to ...
2
votes
0answers
232 views

Geodesic Flow is the flow of the Hamiltonian Vector field of $|\xi|$

Let $M$ be a complete Riemannian manifold with metric $g$. The geodesic flow is the one-parameter family of diffeomorphisms $\phi_t$ on $T^*M$ defined as follows. If $\xi \in T^* M$ is a vector based ...