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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299 views

Is length adimensional when space is not flat?

Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...
6
votes
1answer
495 views

good problem book in differential geometry

What are the books in Differential Geometry with good collection of problems. At present I am having John M.Lee's Riemannian Manifolds,Kobayashi Nomizu's Foundations of Differential Geometry. I ...
5
votes
2answers
140 views

Equivalence of intrinsic and extrinsic metrics of embedded manifolds.

Say a compact n-manifold $\mathcal{M}$ is embedded in $\mathbb{R}^m$, $m > n$. If $d_{\mathcal{M}}$ is the geodesic distance on $\mathcal{M}$, and $d$ the Euclidean distance in $\mathbb{R}^m$, ...
3
votes
0answers
140 views

Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?

I have previously asked about Weitzenböck identities and received some great answers on MathOverflow. One question which has arisen from the post is the following: Let $E$ be a hermitian ...
1
vote
0answers
68 views

Solving to get free falling coordinate as function of arbitrary coordinate

From weinberg's gravitation, EQ : $3.2.11$ $$\frac{\partial^2 \zeta^\alpha}{\partial x^\mu \partial x^\nu} = \Gamma^\lambda _{\mu \nu}\frac{\partial\zeta^\alpha}{\partial x ^\lambda}$$ The solution ...
12
votes
1answer
355 views

Dolbeault Cohomology is invariant under homeomorphisms

If $X$ and $Y$ are two complex manifolds, which are homeomorphic but not necessarily diffeomorphic, must their Dolbeault cohomology groups be isomorphic? Here the Dolbeault cohomology groups ...
4
votes
1answer
239 views

Covariant derivative of a pushforward

Suppose that $\Phi_t$ is a the global flow associated with a vector field $X$ on a Riemannian manifold $M$ and that $Y$ is any other vector field. Suppose furthermore that $X$ is a Killing vector ...
2
votes
0answers
87 views

a problem of comparison geometry: Riemannian manifold with upper bounded sectional curvaturee

I have a question about Comparison Geometry: I have a Riemannian Manifold $(X,g)$, complete and simply connected, with sectional curvature upper bounded by a positive constant $k>0$, so I can ...
2
votes
0answers
180 views

Correct use of substitution rule for Integration on Riemannian manifolds

Let $(N,g_N)$ be a Riemannian manifold and let $\psi: M \rightarrow N$ be a a diffeomorphism. Now I know how the Riemannian metric on $M$ defined by the pull-back of the metric on $N$ looks like (this ...
3
votes
0answers
52 views

Holomorphic analogue of geodesics

Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C ...
1
vote
0answers
103 views

Volume of a ball intersected with a submanifold

I am given a smooth Riemannian submanifold of dimension $d$ embedded in $\mathbb{R}^D$ with condition number $1/\tau$ (a formal definition of condition number is on Page 3 of this paper ...
2
votes
1answer
131 views

notation question - vector field and function on manifold

So I'm trying to learn Riemannian geometry on my own... probably not a realistic goal! But anyway, for now I'm stuck on understanding part of this passage: A vector field $X$ on a $C^{\infty}$ ...
5
votes
2answers
169 views

Tensor Components

I would like to ask something On the Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component: Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset M$. If $A ...
2
votes
1answer
90 views

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 ...
3
votes
1answer
260 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 ...
6
votes
1answer
112 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 ...
1
vote
1answer
77 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 ...
1
vote
1answer
235 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?
6
votes
2answers
403 views

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 ...
5
votes
1answer
293 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 ...
2
votes
0answers
66 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 ...
13
votes
2answers
399 views

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 ...
26
votes
2answers
963 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 ...
4
votes
2answers
119 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 ...
5
votes
3answers
2k views

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 ...
7
votes
1answer
536 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, ...
1
vote
1answer
455 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 ...
3
votes
1answer
148 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 ...
5
votes
2answers
236 views

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. ...
1
vote
1answer
129 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 ...
5
votes
2answers
571 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 ...
0
votes
1answer
883 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
98 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
80 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
49 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
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
votes
1answer
203 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
141 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
96 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
308 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
186 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
37 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
95 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
310 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
149 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
78 views

Curvature on topological spaces

On what subsets of the category of topological spaces are different notions of curvature defined?
3
votes
2answers
139 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
268 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
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 ...
3
votes
0answers
105 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 ...