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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Nonexistence of local isometry between equidimensional Riemannian manifolds

Recall that all inner product spaces of the same dimension are isometric. For example, if $(M,\mathrm{g})$ and $(N,\mathrm{h})$ are Riemannian manifolds of the same dimension, then ...
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1answer
112 views

Injectivity of the Differential of Smooth Map

I am trying to answer the following question: Let $M = \{(x,y)\in \mathbf{R}^2 : x^2 + y^2 < 1\}$. Define a smooth or $C^\infty$ function by $f\colon M \rightarrow \mathbf{R}^2$ as ...
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0answers
54 views

Kinds of isometries preserving the curvature tensor

We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature. First I am ...
4
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2answers
133 views

What's my mistake in the calculation?

Summation convention holds. If $\frac{\partial}{\partial t}g_{ij}=\frac{2}{n}rg_{ij}-2R_{ij}$, then ,I compute: $$ \frac{1}{2}g^{ij}\frac{\partial}{\partial ...
3
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2answers
121 views

Boundedness of the norm of the Riemann curvature tensor

Let $(M,g)$ be a Riemannian manifold and let $R(X,Y)Z$ be its $(3,1)$ Riemann curvature tensor given by $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ Let the input vectors $X,Y,Z$ ...
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1answer
58 views

Invariance of volume along Ricci flow

The Riemannian metric is $g_{ij}$,its inverse is $g^{ij}$,and the induced measure is $du=u(x)du$ where $u(x)=\sqrt{det(g_{ij})}$.The scalar curvature is $R=g^{ij}R_{ij}$ . $r=\frac{\int R du}{\int ...
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2answers
138 views

Statement about the isometries of a product manifold

I'm studying Minkowski spacetime $\Bbb{M}$, and I would like to make the following statement about its symmetry transformations. Since $\Bbb{M}$ is the product manifold of time and space, it inherits ...
2
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2answers
76 views

Geodesics on a generalized cylinder

I want to prove that given a generalized cylinder $C(s,t)=\alpha(s)+t\hat{z}$ , where $\alpha$ is a curve on the $xy$ plane and $\hat{z}$ is the $z$-axis vector, then a geodesic curve $\gamma$ has the ...
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1answer
22 views

Orthogonal Jacobi Fields Remain Orthogonal

Let $\gamma(t)$ be a geodesic and suppose $\left<J(0), \gamma'(0)\right> = \left<J'(0), \gamma'(0)\right> = 0$ where $J'(0)$ indicates the covariant derivative of $J$ along $\gamma$. There ...
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0answers
59 views

“Commutation” of parallel transport with covariant derivative and Riemann curvature tensor

I'd appreciate it a lot of you could please give me a detailed answer to this question. Alternately, you could just cite a reference too. Let $(M,g)$ be a Riemannian manifold, $c$ a curve on it. You ...
2
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2answers
368 views

why does Lie bracket of two coordinate vector fields always vanish?

This is really puzzling me. Say we are dealing with a Riemannian manifold $(M,g)$. Suppose $\nabla$ is the unique torsion free connection on $M$ that is compatible with $g$. Suppose we are in a ...
2
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0answers
89 views

Check Riemannian manifold's isometry to $\Bbb{R}^n$

Let $\mathcal{M}$ be the convex cone of symmetric positive definite $n\times n$ real matrices. $\mathcal{M}$ is an $\frac{n(n+1)}{2}$-dimenasional Riemannian manifold. Could you help me proving (or ...
2
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2answers
148 views

Parallel transport of a vector in hyperbolic space, specifically in $\mathbb{H}$

Let us consider Poincaré's upper plane which is defined as $\mathbb{H} = \{ (x,y) | y>0\}$. This space has a Riemannian metric $g = \text{diag}(1/y^2, 1/y^2)$. Now let us consider a differential ...
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1answer
19 views

Gauss lemma tangent space identification

In Gauss' lemma (in riemannian geometry), many books say that we can identify $T_v(T_xM)$ with $T_xM$ where $(x,v) \in TM$. How can I see how to identify these two spaces?
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35 views

decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g-1)$ pants bounded by 3 geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
3
votes
1answer
72 views

Metric on Homogeneous Space $G/H$

For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where ...
2
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0answers
79 views

Is it possible to put a Ricci-flat metric on the $n$-sphere for $ n>4$?

I'm looking for references which discuss the possibility of putting a Ricci-flat metric on the $n$-sphere for $n > 4$. Thank you for any kind of help.
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1answer
137 views

Totally geodesic hypersurface in compact hyperbolic manifold

In [Zeghib: Laminations et hypersurfaces géodésiques des variétés hyperboliques, Annales scientifiques de l'ENS, 1991] it is shown, that in a compact manifold of negative curvature, there exists only ...
2
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0answers
79 views

Show the negative-definiteness of a squared Riemannian metric

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices. The geodesic distance between $A,B\in\Bbb{S}_{++}^n$ is given by the following Riemannian metric ...
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39 views

Construct a SPD kernel using a (true) distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$ be a non-empty set of pairs $(\mathbf{x},\Sigma_x)$, where $\mathbf{x}\in\Bbb{R}^n$, $\Sigma_x\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes ...
4
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1answer
73 views

A lift of isometry to universal covering

Let $M$ be a compact Riemannian manifold, $\bar M \to M$ be its universal covering and $\phi \in Isom(M)$ be an isometry of $M$. Is it true that, if $\phi$ is isotopic to the identity map of $M$, than ...
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1answer
49 views

Slight confusion about Riemann curvature, in specific about $\nabla_{[X,Y]}$

In what follows I always use Einstein summation convention. The Riemann curvature is defined as $$ R(X,Y)Z = \nabla_{X}\nabla_{Y}Z - \nabla_{Y}\nabla_{X}Z - \nabla_{[X,Y]}Z $$ Now, I want to ...
5
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1answer
116 views

Theorema egregium violated in dimension $n \ge 4$?

Gauß showed that for surfaces in $\mathbb{R}^3$ the Gaussian curvature ( = sectional curvature) is invariant under local isometries. This is known as the thema egregium. Now in another question ...
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1answer
90 views

Meaning of “locally homeomorphic to $\mathbb{R}^{n}$”

I am fairly new to differential geometry and approaching it with a physics background (in the study of general relativity), as a result I'm having a few struggles with terminology etc, so please bear ...
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44 views

Does the coarea formula hold for smooth maps with gradient bounded below?

The coarea formula for hypersurfaces in $\mathbb R^n$ can be written in two following forms: $$ \int_{\mathbb R^n} g(x) |\nabla u(y)| dx = \int_{\mathbb R} \int_{u^{-1}(t)} g(y) d\mathscr H^{n-1}(y) ...
4
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1answer
84 views

Why is this map $H^1$?

I have the following proposition (taken from Klingenberg's Lectures on Closed Geodesics): Let $\pi: E \rightarrow S$ and $\mathcal{O} \subset E$ be a finite dimensional fibre bundle over the ...
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1answer
52 views

“measure zero” and “measurable function” on Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold (which doesn't have to be orientable). As far as I know, the metric $g$ induces a "canonical" measure $\mu$ and so one can talk about sets $U\subset M$ of measure ...
2
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0answers
40 views

All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible ...
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19 views

characeterization of zero sets of the riemannian measure of a riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold (does not have to be orientiable). Then there exists the Riemannian measure $\nu(g)$ on $M$. Let $(U_i,x_i)$ be a finite covering of $M$ of charts and let ...
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26 views

Bounding distance between geodesics in manifolds with nonpositive curvature

I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I would like to see proven (and clarified). Let $M$ be a compact, connected ...
2
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1answer
96 views

Lagrange's Equation on a Manifold

I know that, if $L: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}$, then the Euler-Lagrange equation is: $$ \nabla_x L - (\nabla_{\dot{x}}L)' \equiv 0$$ In trying to ...
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1answer
34 views

Mean Curvature Flow equation, where does it come from?

We consider a compact, uniformally convex, $n$-dimensional surface $M=M_0$ without boundary imbedded in $\mathbb{R}^n$. We want to find a family of maps satisfying the evolution equation ...
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3answers
129 views

Proving smoothness of left-invariant metric on a Lie Group

Assume $G$ is a Lie group. The standard construction of a left invariant metric on $G$ goes as follows: Take an arbitrary inner product $\langle,\rangle_e$ on $T_eG$ and define $\langle u , ...
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1answer
45 views

Tensor manipulation

I am very new at manipulating tensors and I have the following equation: $$A_{\mu \nu\tau} b^\mu c^\nu = g_{\tau \rho} d^\rho$$ where $\tau$ is the independent index and $g_{\tau \rho}$ the metric ...
2
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0answers
45 views

How to Induce a Metric on Homogeneous Space $G/H$ by the Metric from Bundle G

I am having a question on how to induce a metric $g$ on homogeneous space $G/H$, if one is given a ${\rm Ad}_H$-invariant metric $\bar{g}$ on G. More specifically and simply, consider principal ...
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1answer
44 views

Potential of metric tensor

As I understand so far, the metric tensor of a Riemannian manifold is an $n \times n$ matrix in many specific examples. As such it could formally be the curl of some vector potential or just the ...
5
votes
2answers
101 views

Unique metric for the Hyperbolic Half Plane Model?

I was reading today that there is a unique metric (up to multiplicative constant) that preserves distances wrt to linear fractional transformations: $$z \mapsto \frac{az + b}{cz + d}$$ of the upper ...
2
votes
1answer
66 views

Normal coordinate parallel along radial geodesics?

A radial geodesic in normal coordinates is given by $\gamma:t \mapsto t(V_1,....,V_n).$ Is it then true that any normal coordinate $\partial_x|_{\gamma}$ is parallel along $\gamma,$ i.e. ...
3
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0answers
82 views

Doubts about a proof by Petersen regarding totally convex sets

I have several doubts about the proof of Lemma 62 in Petersen's Riemannian Geometry book (pp. 355-356 in the second edition). Why is $f\le d(\cdot,\partial A)$? One could argue that a segment from ...
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1answer
68 views

parallel vector field

I was wondering about the following: I know that a vector field along a geodesic that is parallel has a constant angle to the tangent vector of the curve and constant length. Now, is the converse ...
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2answers
132 views

Why the flat torus cannot be immersed in euclidean plane?

I am trying to prove the following claim: The flat $2$-dimensional torus cannot be isometrically immersed into $\mathbb{R}^2$ with the standard metric. That is, there is no immersion $f:T^2 ...
2
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1answer
40 views

Geodesic flow on a compact manifold is defined for all time

How can I prove that on a compact manifold, the geodesic flow is defined for all time? Is this as simple as citing the Hopf-Rinow theorem?
2
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1answer
77 views

Geodesic Flow is an Anosov Flow

I am trying to understand why geodesic flow on a compact surface of constant negative curvature is an Anosov flow. Klingenberg's book, Riemannian Geometry, says that in this case, the proof is very ...
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2answers
106 views

Is it mathematically correct to say that if the metric is flat/curved the *shortest* path is/not a Euclidean straight line?

Is it mathematically correct to say that if the metric is flat/curved the shortest path is/not a Euclidean straight line? I am still hesitant to make this claim, due to at least one counter example. ...
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2answers
109 views

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = ...
3
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1answer
62 views

Mean Curvature Flow

Recently I am reading the mean curvature flow from the lecture notes of Carlo Mantegazza where I found that Under mean curvature flow given by$$\begin{cases}{\partial\over \partial ...
5
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1answer
75 views

Divergence of a tensor with respect to the Levi-Civita connection

In a Riemannian manifold $\mathcal{S}$ with metric $\boldsymbol{g}$, given a chart $\{x^a\}$, it is fairly easy to prove that the divergence of a vector field $\boldsymbol{w} : \mathcal{S} \to ...
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1answer
67 views

(Co)Tangent bundle of Cone manifold

Given a Riemannian manifold $(M,\bar{g})$, we can construct the Riemannian cone manifold $(C(M), g )$ as follows. Topologically, $C(M)$ is $M \times \mathbb{R}_{>0}$. We equip this with the ...
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1answer
27 views

Show that the section $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$

Show that the section $g$ of $T^*\mathbb{R}^3 \otimes T^*\mathbb{R}^3$ defined by $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$ and compute ...
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0answers
100 views

use of existence of bi-invariant differential form on a Lie group?

In do-carmo's Book "Riemannian Geometry" there is an exercise on proving existence of a bi-invariant metric on any compact connected Lie group. (pg 46, question 7). In the first stage, you are ...