(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.

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167 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 ...
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0answers
113 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 ...
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84 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
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1answer
277 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
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1answer
149 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?
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33 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 ...
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1answer
86 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
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1answer
275 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
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1answer
120 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 ...
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71 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
3
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2answers
111 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 ...
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1answer
177 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 ...
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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
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0answers
91 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 ...
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2answers
101 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
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1answer
194 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
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3answers
290 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
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1answer
286 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 ...
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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 ...
3
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0answers
120 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
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1answer
225 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
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2answers
304 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 $d(p_0,N)$ ...
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2answers
233 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 ...
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0answers
55 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 ...
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1answer
299 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 ...
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69 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, ...
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136 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 ...
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2answers
965 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 ...
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1answer
54 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 $r<R$, ...
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1answer
220 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 ...
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2answers
40 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
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1answer
185 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 ...
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2answers
364 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 $\Delta_{M\times N}$ ...
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1answer
125 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 ...
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641 views

A conformal map from a horizontal half-strip to $H$

I have seen many examples of mapping the vertical half-strip, ie $-\pi/2 \lt x < \pi/2$, $y \gt 0$ to $H$(the upper half-plane) in $\mathbb{C}$ using the transformation $f = \sin z$. Would the ...
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0answers
42 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 ...
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1answer
122 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 $P_s$ ...
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1answer
261 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 ...
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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 ...
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0answers
51 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
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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, ...
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1answer
46 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 ...
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1answer
179 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 ...
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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
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1answer
88 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 ...
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209 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 ...
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2answers
258 views

An exercise in the Riemannian geometry book

If $M$ is a smooth closed $n$-dimensional Riemannian manifold which is Riemannian embedded in $\mathbb R^{n+1}$, then there exists a point $p \in M$ such that the sectional curvatures at $p$ are all ...
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1answer
89 views

The immersion of Riemannian manifold

Assume that we have a Riemannian immersion of an $n$-dimensional Riemannian manifold into $\mathbb R^{n+1}$. If $n ≥ 3$, then does $M$ necessarily have non-negative sectional curvature?(I know if ...
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1answer
98 views

The degree of Gauss map

If $M$ is an $2m$-dimensional closed orientable hypersurface in $\mathbb R^{2m+1}$, then we have a Gauss map $G:M\rightarrow S^{2m}$. I have known from my differential geometry book that ...
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1answer
211 views

Bounding the Norm of the Riemann Curvature Operator

I am having trouble with exercise 26 in chapter 2 of Peter Petersen's text "Riemannian Geometry." The exercise is stated: "Using Polarization show that the norm of the curvature operator on ...