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 ...

learn more… | top users | synonyms

2
votes
1answer
106 views

Calculus on manifolds

I am studying manifold theory ( smooth manifold). As I study the subject, I feel more and more confused. I am not yet at chapter dealing with deferential forms and integration. I can understand that ...
2
votes
2answers
80 views

Nonlinear PDE from Riemannian Geometry

I am wondering if anyone knows an approach to finding solutions to the following PDE: $-e^{-2u}\Delta u=\alpha$. Here $u=u(x,y)$ is an unknown real-valued function of 2 variables and $\alpha$ is a ...
1
vote
0answers
183 views

Geodesic question

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in ...
1
vote
0answers
59 views

Connection on $\operatorname{Spin}^\mathbb{C}$ spinor bundle

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus ...
1
vote
2answers
68 views

Forms on Riemann Surfaces

I want to show that the space of smooth $(1,0)$ forms on a compact Riemann surface $X$ has the natural splitting: $\mathcal{E}^{1,0}(X)=\Omega(X) \oplus \partial\mathcal{E}^{0}(X)$, where $\Omega(X)$ ...
2
votes
1answer
123 views

Gradient Ricci soliton

I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons". A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth ...
3
votes
1answer
129 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 ...
1
vote
1answer
130 views

Autoparallel submanifolds and geodesics

I have the following question in differential geometry. Any help is greatly appreciated. Let $M$ be an autoparallel submanifold of a manifold $S$ with respect to a connection $\nabla$. Let $\gamma$ be ...
3
votes
2answers
126 views

How do geodesics change when I scale the metric?

If (M,g) is a Riemannian manifold, and f(m) is a positive real-valued function on M, then f.g is another Riemannian metric on M. If I know all the g-geodesics from x to y in M, can I find out the ...
1
vote
2answers
171 views

Why do people stick with Riemann-Integration when dealing with differential geometry?

I asked a question yesterday that is, "Is there an introductory differential geometry text using Lebesgue integration?" Then, i got an answer that "since we are dealing with differential geometry we ...
1
vote
1answer
262 views

Metric and Curvature on a Riemann Surface

We are given a smooth conformal metric $\rho=\rho(z)\left|dz\right|$ on a Riemann surface $X$. I have a few questions relating to this: (a) The local formula $R(\rho)=\Delta \mathrm{log}\rho dx\,dy$ ...
3
votes
0answers
134 views

Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
10
votes
1answer
188 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
10
votes
1answer
131 views

Using index notation to write $d^2=0$ in terms of a torsion free connection.

Let $(M,g)$ be a Riemannian manifold and let $\omega$ be a $1$-form on $M$. I want to rewrite $d^2\omega=0$ in terms of the Levi-Civita connection. I can show the following: $$d\omega(X,Y) = ...
4
votes
0answers
141 views

$\operatorname{div}$ and $\operatorname{grad}$ in spherical coordinates. Formula from general relativity goes crazy

I try to calculate the gradient of a function and the divergence of a vector field in spherical coordinates. Nothing special so far, but a formula that I learned in a general relativity lecture ...
1
vote
1answer
57 views

About Whitney Theorem

Note that $M$ of dimension $n$ can be imbedded differentiably as a closed submanifold of ${\bf R}^{N=2n+1}$. Here Let $f$ be an imbedding. $f$ is one-to-one immersion, that is, rank $n$, which is ...
3
votes
1answer
105 views

Flatness of a manifold (or a connection)

Suppose we have an $n$-dimensional manifold $S$ (with a global coordinate system) with a metric $g$ and a connection $\nabla$ with connection coefficients (Christoffel symbols) $\Gamma_{i,j}^k$ given. ...
4
votes
1answer
94 views

surface measure under induced surface metric

I'm currently reading a paper about incompressible Euler's equation, and I don't understand how the surface element expand. So here comes the question. Let $\Omega$ be a Riemannian manifold with ...
0
votes
1answer
57 views

hamiltonian mechanics

In $\mathbb{R}^{2n}$, $\omega=\sum dx_i \wedge dy_i$ is a canonical symplectic form, and H is an hamiltonian function, i.e. $\dot{x}= \frac{\partial H}{\partial y}$, $\dot{y}= -\frac{\partial ...
0
votes
1answer
51 views

The distance in Riemann manifold

Let $f: M\to M$, where $(M,\rho)$ is a closed Riemann manifold, and $(\widetilde{M},\widetilde{\rho})$ is the universal covering of $(M,\rho)$, $D$ is a fundamental domain of ...
2
votes
1answer
145 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
0answers
38 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
0
votes
2answers
67 views

Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
1
vote
2answers
108 views

Is the Laplacian $-\Delta$ on a compact manifold an isomorphism?

We know that for (a normal) domain $-\Delta:H^1_0(\Omega) \to H^{-1}(\Omega)$ is an isomorphism. What is the corresponding result for the Laplace-Bulltrami operator or more generally a Laplacian ...
1
vote
0answers
169 views

Isometry group of a compact manifold

Is an isometry group of a compact manifold always a compact group?
1
vote
1answer
69 views

Parallel translation via $e$-connection

This question is concerned with Section 2.5. of Amari and Nagaoka's Information geometry book. Let me give some background first. Let $\mathcal{P}$ be the $n$-dimensional manifold of all (strictly ...
0
votes
1answer
40 views

1-form on the Riemannian manifold

Let $\omega$ a 1-form on a riemannian manifold $(M,g)$, and for a point $x\in M$, there is a notation: $|\omega_x|_g$, what does $|\omega_x|_g$ mean?
3
votes
1answer
252 views

Show isometry of flow on a compact Riemannian manifold where the vector field is Killing

Let $(M,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection of $g$. A vector filed $V$ on $M$ is called a Killing field if for every $p\in M$ and every $X,Y\in T_p M$, $$ g(\nabla_X V, ...
1
vote
1answer
83 views

Find the Gauss Curvature of This Particular Metric:

Let D be an open disc centred at the origin in $ \Bbb R^2 $. Give D a Riemannian metric of the form $ (dx^2 + dy^2)/f(r)^2 $, where $ r = \sqrt{x^2 + y^2} $ and $ f(r) > 0 $. Show that the Gauss ...
0
votes
0answers
83 views

The trace of a wedge product of matrices

I'm trying understand a computation on Besse's book (p. 371). I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form relative to the direct sum ...
4
votes
1answer
105 views

Is this a geodesic?

Let $(M,g)$ be a riemannian manifold. Let $p$ in $M$ and $v,v_{0}$ two vectors in $\mathrm{T}_{p}M$. I am looking at the curve $$ \gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0}) $$ ...
1
vote
1answer
136 views

Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
1
vote
0answers
52 views

Notations in Riemannian Geometry

Let $f:M\rightarrow N$ be differential map. We denote tangent map $$f_*:TM\rightarrow TN$$ and cotangent map $$f^*:T^*N\rightarrow T^*M$$ Now let $M$, $N$ be Riemannian manifolds, and ...
0
votes
1answer
100 views

Metric completion of universal covering of punctured plane

It is known that the universal covering of the punctured plane $\mathbb C\setminus\{0\}$ is $\exp:\mathbb C\to\mathbb C\setminus\{0\}$. In real coordinates, $f=\exp:\tilde M=\mathbb R^2\to M=\mathbb ...
0
votes
2answers
103 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
3
votes
1answer
53 views

Isometric map of geodesic

Assume a Riemann manifold $(M,g)$ and a smooth map $\sigma:M\times M\rightarrow M$, $(m_{1},m_{2})\rightarrow \sigma_{m_{1}}(m_{2})$, such that: $\forall m\in M$ $\sigma_{m}:M\rightarrow M$ is an ...
1
vote
0answers
306 views

Chain rule with covariant derivative

Let $\mathcal{M}$ be a $n$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. Consider the following function: $$\tilde{F}(v) = \operatorname{d exp}^{-1}_{p} ...
0
votes
1answer
45 views

precise meaning of connected manifold

what does it mean for a manifold to be "connected" precisely? what is the difference between a connected riemannian manifold and a nonconnected one. (i know what a riemannian manifold is a manifold ...
0
votes
1answer
176 views

connected complete totally geodesic sub manifold of $S^n$

Let $M$ and $N$ be manifolds with Riemannian metrics $g$ and $h$ respectively. A diffeomorphism $F: M\to N$ is an isometry if \begin{equation*} h_{F(x)}(T_x F(u), T_x F(v))=g_x(u,v) ...
0
votes
1answer
76 views

embedding discrete metric into manifold?

True or false: "Any edge-weighted undirected graph can be isometrically embedded into some Riemannian manifold". "isometric embedding" here means that for any pair of nodes, their shortest path ...
0
votes
1answer
110 views

Covariant derivative along curve

Let $M={\mathbb R}^{3} $ with the usual metric $g=ds^{2} =dx^{2} +dy^{2} +dz^{2} $. Let $\gamma :I\to M$ be a unit speed curve. How can I prove that $\nabla _{\gamma '} \gamma '=\gamma ''$ , where ...
1
vote
1answer
55 views

Local coordinates for two riemannian metrics

Let $(M,g)$ be a Riemannian manifold, $g' = g + f$ be another metric. Is it possible to get local coordinates such that at a point $P \in M$, $g_{ij} = \delta_{ij}$ and $f_{ij} = 0$ for all $i \not = ...
0
votes
1answer
144 views

Hamilton's Proof of the Tensor Maximum Principle

So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of Theorem 9.1. I'm ...
2
votes
0answers
428 views

Taylor expansion of a vector field on manifold

In my work I have a need for some kind of analogue of Taylor expansion of a vector field on Riemannian manifold $\mathcal{M}$. I came to such an expression: $$ F(\operatorname{exp}_p(v)) = ...
2
votes
1answer
98 views

Gradient of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how is it then true that $(\text{grad}f|_M)_p$ at a point $p$ (gradient of the mapping ...
6
votes
2answers
287 views

Topology on the space of compatible almost complex structures in symplectic geometry

I have a few fairly generic questions, with a specific application to symplectic geometry in mind. Let me pose the specific problem first: Let a symplectic manifold $(M,\omega)$ be given. One is ...
1
vote
1answer
255 views

Bishop - Gromov Comparison Theorem proof and references.

I'm having trouble understanding a proof of the Bishop's volume comparison theorem and any help would be really appreciated. It's a simple part of the proof but I'm not quite getting what they want to ...
3
votes
1answer
87 views

Hilbert-Schmidt norm/smooth manifolds

Given two riemannian manifolds $M$ and $N$ and a smooth map $f$ : $M$ $\rightarrow$ $N$, we define the energy density of $f$ as the smooth function $e(f)$ : $M$ $\rightarrow$ $\mathbb{R}$ given by ...
0
votes
1answer
67 views

On Vanishing Riemann curvature tensor

If a manifold $\mathcal{M}$ has a vanishing Riemann curvature tensor, then what i) does this imply for the manifold? and ii) What is such a manifold called?
0
votes
1answer
88 views

How does curvature do that?

In his book "Riemannian geometry" Do Carmo said The curvature measures the amount that a riemannian manifold deviates from being euclidean My question is How does the curvature measure this ...