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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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 ...
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56 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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92 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 ...
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72 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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34 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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26 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
41 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
96 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
67 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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18 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) ...
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76 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
31 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 ...
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29 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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15 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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18 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
39 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
21 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
103 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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2answers
28 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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28 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
29 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
87 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
50 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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81 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
44 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 ...
1
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1answer
52 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 ...
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1answer
30 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
29 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 ...
2
votes
2answers
70 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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0answers
56 views

Covariant derivative and box operator commutator

I know that the commutator of two covariant derivatives is giving some Riemann tensors as follow: ...
3
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2answers
64 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
38 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
votes
1answer
32 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
42 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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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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32 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 ...
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Uniqueness of bi-invariant metrics on Lie groups?

As noted here , a Lie group $G$ admits a bi-invariant metric if and only if $G$ is the cartesian product of a compact (Lie) group and a vector space $\mathbb{R}^n$. The question: For which Lie ...
3
votes
2answers
44 views

Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
4
votes
1answer
29 views

Calculate the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...
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1answer
66 views

Curvature and Circumference of Circle

Theorem Let $\gamma\colon [a,b]\rightarrow \mathbb{R}^2$ be a unit speed simple closed curve, with $\gamma'(a)=\gamma'(b)$ and $N$ is the inward-pointing normal. Then $$ \int_{a}^b ...
2
votes
1answer
28 views

An inequality for absolute total curvature in Riemannian surfaces

Let be $M\subseteq \mathbb{R}^3$ a compact (Riemannian) surface and let be $K$ the gaussian curvature of $M$. I want to prove that $$ \int_{M} |K| \geq 4\pi(1+g(M))$$ where $g(M)$ is the genus of ...
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1answer
69 views

Ricci tensor and average of a tensor

Let $(M^n,g)$ be an oriented Riemannian $n$- manifold and $g$ is a Riemannian metric on $M$ , $\mathrm{d}\sigma$ is Riemannian volume form on $S^{n-1}$ and $\text{Vol}(S^{n-1})$ is volume of ...
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30 views

a problem on geometry of hypersurfaces

Recently I am reading book on mean curvature flow by carlo mantegazza.There I found a problem on hypersurfaces stated below : Show that if the hypersurface $M \subset {R}^{n+1}$ is locally the graph ...
2
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1answer
41 views

Isometry in Hyperbolic space

Let $\mathbb{H}^2=\{ (x,y)\in\mathbb{R}|\ y>0 \}$ the hyperbolic space with the metric $g=(dx^2+dy^2)/y^2$. Let ...
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22 views

Two successive isometric immersions: relation between mean curvature vectors?

Let $M_0$ be a Riemannian manifold, $M_1$ a geodesic sphere of $M_0$ and $M_2$ an isometrically immersed submanifold of $M_1$, ie: $$ M_2 \subset M_1 \subset M_0$$ Take $X \in M_2$, and: $T_2$ the ...
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139 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
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35 views

Generalizing Pearson's coefficient to determine properties of embedded manifold

I have the following dilemma: We know that for random vectors we have Pearson's coefficient of skewness. I think you all agree that in some sense it measures the shape properties of the ...
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78 views

Vectors that geodesically generate the same surface

Suppose that $\langle M,g \rangle$ is a complete, simply connected Riemannian symmetric space. The surface geodesically generated by a vector $\xi$ in $T_pM$ is the set of points lying on geodesics ...
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36 views

Space of $G$-invariant Riemannian metrics contractible?

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ ...
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1answer
41 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...