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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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 t}g_{ij}=\frac{1}{2}g^{ij}(\frac{2}{n}rg_{...
3
votes
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 $X,Y,Z$ ...
-1
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1answer
60 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 $R=g^{ij}R_{ij}$ . $r=\frac{\int R du}{\int du}$...
1
vote
2answers
140 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
votes
2answers
77 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 ...
1
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1answer
23 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 ...
1
vote
0answers
62 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
votes
2answers
440 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
90 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
votes
2answers
157 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 ...
1
vote
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?
3
votes
1answer
79 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
votes
0answers
84 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.
8
votes
1answer
142 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
votes
0answers
80 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 ...
4
votes
1answer
87 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 ...
0
votes
1answer
53 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
votes
1answer
119 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 (...
1
vote
1answer
92 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 ...
0
votes
0answers
47 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
votes
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 ...
1
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1answer
63 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
votes
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 one-...
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0answers
29 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
votes
1answer
108 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 ...
1
vote
1answer
37 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 $\frac{\...
2
votes
3answers
137 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 , v\...
0
votes
1answer
46 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
votes
0answers
53 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 ...
1
vote
1answer
46 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
103 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
72 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. $\nabla_{\...
3
votes
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 $...
1
vote
1answer
71 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
vote
2answers
141 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
votes
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
votes
1answer
79 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
109 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. ...
3
votes
2answers
116 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 = \int_{t_1}^{t_2}...
3
votes
1answer
67 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 t}\varphi(p,t)=...
5
votes
1answer
84 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 T\...
0
votes
1answer
68 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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votes
1answer
31 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 the ...
2
votes
0answers
124 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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0answers
151 views

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
67 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
37 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 ...
1
vote
1answer
105 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 \kappa_N(s)ds=2\pi....
2
votes
1answer
45 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 $...
1
vote
1answer
94 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 $S^{n-1}...