A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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Open sets of the tangent bundle in a Riemannian manifold

Let $M$ be a Riemannian manifold with a metric $g$ and $(U,\varphi)$ a chart around a point $p\in M$. By a Remark page 63 of Riemannian Geometry by M. Do Carmo, it seems that any open set ...
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274 views

Exponential map on manifolds and differential

I am trying to understand the proof of Theorem 3.7, page 72 of Riemannian Geometry by M. Do Carmo. For $M$ a Riemannian manifold and $(U,\varphi)$ a chart around a point $p\in M$, he (more or ...
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117 views

Holonomy group of quotient manifold

Let $(M,g_M)$ be a compact Riemannian manifold with holonomy group $Hol(M,g_M)$. Suppose that a finite group $G$ acts on $M$ freely and preserves the metric $g$. What can one say about the holonomy ...
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78 views

in Riemannian geometry, when is there an ambient space?

I am reading Kuhnel's Differential Geometry of Curves,Surfaces,Manifolds (2ed). On p.209, discussing tangent space of riemannian manifold, it says: ``since there is no ambient space, this notion has ...
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266 views

Isometry and harmonic forms

Let $M$ be a Riemannian manifold. Assume that a finite group $G$ acts on $M$ as isometry. How can one prove that $G$ takes harmonic forms to harmonic forms?
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do Carmo: Second Variation Formula

I have some trouble with the derivation of the second variation formula in do Carmo's famous "Riemannian Geometry" (p. 197f.). The proposition is the following: 2.8 Proposition Let ...
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329 views

A vector bundle admits a local covariantly constant section iff it is flat

Let $p:E\rightarrow M$ be a vector bundle over a manifold $M$ and let $\nabla$ be a connection on $E$. I am trying to show that $E$ admits a covariantly constant section $s$ in a neighborhood of each ...
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67 views

Convex polyhedral decomposition of spheres

Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$? Remarks: A polyhedron is defined as an area enclosed by a ...
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metric in the Wasserstein space of gaussian measures

I am reading the paper "Wasserstein Geometry of Gaussian measures" by Asuka Takatsu (section 3 is of interest to me) and I have difficulties understanding how the metric is used. In particular, I am ...
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1k views

PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality ...
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126 views

terminology question - exponential map

The exponential map goes from the tangent space to the manifold, and the log map goes back. In reading, however, I get the impression that people use the "exponential map" as a term for the overall ...
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Riemannian/Ricci curvature for round n-sphere

What is the best way to see that the Ricci scalar curvature of $(S^n(r),g_{round})$ is a constant $n(n-1)/r^2$ ? I essentially only see this value stated in the literature, but no computation ...
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559 views

Directional derivative of vector field

I am trying to compute the directional derivative of a vector field $V$ along a direction $U$. Actually, my vector field is initially only defined on a curve $\gamma(t)$ in a Riemannian manifold $(M, ...
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1answer
492 views

Commutators, and Christoffel symbols in a non holonomic basis

I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors. I need to compute ...
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1answer
149 views

Derivative of a parallel translation inside a metric

Let $M$ be a riemannian manifold with metric $g$ and a connection $\nabla$ on $M$. Let $X,Y$ two vector fields along a curve $\gamma$ on $M$. Let $$\tau_{t,s}:T_{\gamma(s)}M\to ...
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Is there an “opposite” of a geodesic?

If I understand correctly, a geodesic between two points $a$ and $b$ is the "most direct" path from $a$ to $b$. Geodesics on a plane are straight lines, geodesics on a sphere are great circle arcs. ...
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1answer
132 views

Evaluating the second fundamental form for a curve

I am (numerically) computing the second fundamental form for a curve $\gamma(t)$ embedded in a Riemannian manifold $(M, g)$. I would like to double check if what I am doing is correct. First, define ...
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609 views

Geodesics on the product of manifolds

Given two Riemannian manifolds $(M, g_1)$ and $(N, g_2)$, and geodesic curves $\gamma(t)$ in $M$ and $\chi(t)$ in $N$. Is the curve $\Gamma(t) = (\gamma(t),\chi(t))$ a geodesic in the product manifold ...
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964 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 ...
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101 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 $T_x M$ ...
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81 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 ...
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51 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 ...
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123 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
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1answer
211 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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147 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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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 ...
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322 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, ...
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209 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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38 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
97 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
324 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 ...
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1answer
155 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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80 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
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145 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
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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
24 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 ...
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112 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
155 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
211 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
396 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 ...
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1answer
455 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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60 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 ...
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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 ...
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285 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 ...
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496 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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312 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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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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464 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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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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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 ...