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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Isometry group of a Lie group

I'm having trouble dealing with the following question : what is the isomety group of $\mathbf{PSL}_2(\mathbb{R})$ viewed as a Lie group with its Killing form ? For the record, its Killing form is the ...
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1answer
140 views

great circle distance

in the euclidean plane the distance from the origin to a point is $s^2 = x^2 + y^2 $ I am reading a paper which say that this could be called an algabraic metric for the plane. the paper then ...
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1answer
89 views

Question about curvatures of hypersurfaces

Let $M^n\subseteq {\mathbb{R}}^{n+1}$ be a hypersurface. Compute the sectional curvatures in all planes which are spanned by two eigenvectors $X_i, X_j$ of the Weingarten map. Also compute the Ricci ...
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1answer
285 views

A question about left invariant vector fields

Let $G$ be a Lie group with bi-invariant metric $\langle , \rangle$ and $X,Y,Z$ left invariant vector fields in $G$, how to conclude that $X\langle Y,Z\rangle=0$?
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1answer
224 views

Definition of Sectional Curvature

do Carmo gives a definition of sectional curvature as follows: $$K(x,y) = \frac{\langle R(x,y)x,y\rangle}{|x\times y|^2}$$ where $x,y \in T_pM$ are linearly independent vectors. My question: The ...
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1answer
81 views

Two Questions about Sobolev inequalities and Lipschitz smooth functions

Question1 I want to find a manifold such that the Sobolev inequality on M of the form $\lVert f \rVert_{n/(n-1)} \leq C\lVert \bigtriangledown f \rVert_1$, where $C=C(n)$, implies that $vol(B(r)) ...
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2answers
217 views

positive non-constant harmonic function $f $ in $L^1(M)$ on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, does there exists a positive non-constant harmonic function $f \in L^1(M)$? Who can answer me or give me a counter example? Thank you very much!
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4answers
824 views

Why are we interested in closed geodesics?

There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds). In the case of geodesics ...
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1answer
248 views

Meaning of the Soul Theorem

The Soul Theorem states that in every complete, connected riemannian manifold $M$ with $\mathrm{sec}(M)\geq 0$, there exists compact, totally convex, totally geodesic submanifold $S$ such that $M$ is ...
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1answer
553 views

Sufficient Conditions for Ricci Tensor to be Diagonal

What are the strongest (or most useful) conditions on a metric for it's Ricci tensor to be diagonal? I've read that if the metric is explicitly dependent on only one variable then the Ricci Tensor is ...
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1answer
74 views

Variations in a Riemannian Manifold

Let be $M$ a Riemannian manifold and $X,Y$ vector fields over $M.$ Now take $p\in M$ arbitrarily, my question is, how construc a variation $f:U\to M,$ $$U\subset ...
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1answer
167 views

Does the value of the covariant derivative at a point of the metric tensor depend only on the involved tangent vectors?

Let $\nabla$ be an affine connection on a pseudo-Riemannian manifold $(M,g)$. Let $c:[0,1] \rightarrow M$ be a differentiable curve and consider vector fields $Y,Z$ along $c$. Is it true that the ...
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1answer
153 views

Defintion of totally geodesic flat submanifold

I don't know if this is an inappropriate question to post on stackexchange, but could somebody give me (reference me) a precise definition of "totally geodesic and flat submanifold" of a riemannian ...
3
votes
2answers
424 views

An example that a Riemannian manifold that is complete, non compact and has finite volume.

I can't think of such an example, which is a complete, non compact Riemannian manifold and has finite volume.
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2answers
184 views

Tangential component of vector field

I'd like to know the definition of "tangential component" in this case, it is question 3 of page 57 of Do Carmo's book Riemannian Geometry: It says: Define $\nabla_XY(p) = $ tangential component ...
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1answer
51 views

Equivalence of two definitions of path (in $\mathbb{R}^3$) length

In a previews question I asked here I used the following definition of path length:$\gamma=(x(t),y(t),z(t))$ : $L(\gamma)=\intop_{a}^{b}\sqrt{(x'(t))^{2}+(y'(t))^{2}+(z'(t))^{2}}$. In the answer ...
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2answers
86 views

How to show that if $P_1,P_2\in\mathbb{R}^3$ then the straight line connecting them is the shortest one?

Let $P_1,P_2\in\mathbb{R}^3$ and consider all the paths from $P_1$ to $P_2$, I wish to prove that the euclidean distance (that is the length of the line connecting them) is the distanse of the ...
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1answer
136 views

Compatible connection over a riemannian manifold

How do I prove the following assertion: Let $\nabla$ be a connection on a riemannian manifold. $\nabla$ is compatible with the metric if and only if for all vector fields $X,Y,Z$ we must have: ...
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1answer
359 views

First Variation Formula

I have a riemannian manifold $M$ and a smooth curve $\alpha$. I want to take a variation of $\alpha$ and apply the first variation formula of arc length but I want to know if it is possible to take ...
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1answer
435 views

The conditions of a metric to be geodesically complete

On $\{\vec{x}\in \mathbb{R^n}:x_1^2+x_2^2+\cdots+x_n^2<1\}$ in $\mathbb {R^n},$ what $\alpha$ can make the metric $$g=(1-x_1^2-x_2^2-\cdots-x_n^2)^{-\alpha}(dx_1\otimes dx_1+dx_2\otimes ...
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1answer
346 views

Most succinct proof of the uniqueness and existence of the Levi-Civita connection.

Seeing as proving the existence and/or uniqueness of the Levi-Civita connection seems to crop up in every single exam in Geometry and General Relativity, what is the most succinct proof of this, to ...
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2answers
486 views

Riemann tensor on a sphere

I was given this exercise but I don't even know where to start: to compute the Riemann tensor of the 2-dimensional sphere. The tensor acts on vector fields X,Y,Z like this: ...
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1answer
183 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
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1answer
105 views

A question about covariant derivative

Let $V,W$ smooth vector fields along a smooth curve $c:I \rightarrow M$ , where $M$ is a Riemannian manifold, if $\frac{d<V,W>}{dt}=0$ why we must have $<V,W>=$ constant?
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1answer
67 views

$\operatorname{Isom}{(M)}$ has Lie-structure for M metrizable manifold

Suppose $M$ is a smooth and metrizable manifold. Then $\operatorname{Isom}{(M)}$ can be given the structure of a Lie group, so that the action of $\operatorname{Isom}{(M)}$ on $M$ is still smooth. I ...
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239 views

How to prove the holonomy group is preserved under Ricci flow?

I've heard that on a Kähler manifold $(M,g_0)$, if you evolve the metric $g$ by Ricci flow $\partial g_{ij}(t)/\partial t=-2R_{ij}$, and $g(0)=g_0$, then you always have $g(t)$ is a Kähler metric on ...
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votes
1answer
635 views

Laplace-Beltrami operator for curves

I'm CS major and have used discrete Laplace-Beltrami operator for 2D-manifold (surface meshes). I'm wondering if it is possible to define Laplace-Beltrami operator for 1D-manifold. If this is ...
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1answer
332 views

is the geodesic flow on Hyperbolic Plane completely integrable?

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and ...
4
votes
2answers
782 views

Riemannian metric in the projective space

Let $A: \mathbb{S}^n \rightarrow \mathbb{S}^n$ be the antipode map ($A(p)=-p$) it is easy to see that $A$ is a isometry, how to use this fact to induce a riemannian metric in the projective space such ...
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votes
1answer
56 views

Upper bound on volume growth

If the Ricci curvature of a compact Riemannian manifold of demsnion $n$ is greater than 1-n, does it follow that the volume entropy satisfies $$\liminf_{r\rightarrow \infty} \frac{\log vol ...
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votes
1answer
169 views

Calculating Geodesics for submanifolds

I am trying to become acquainted with the notion of geodesics. When we consider a Submanifold $M\subset \mathbb{R}^n$ and a curve $c:I\rightarrow M$. Now I want to know how to check whether c is a ...
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1answer
263 views

Some results about Levi-Civita connection in euclidean space?

I calculated some properties of the Levi-Civita Connection on a semi-riemannian manifold. But I'm not sure, whether my results are correct. Can you please tell me, when something below is wrong or if ...
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1answer
131 views

A new metric involving curves

Let $(X, d)$ be a metric space. The inner metric or length metric associated with $d$ is the function $d_i : X \times X \to [0,\infty]$ defined by $$d_i(x, y) := \inf L(\sigma)$$ where the infimum is ...
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1answer
108 views

Question about immersions of $\mathbb{R}P^n$ into $\mathbb{R}^{n+1}$

I am currently reading a paper which takes for granted the following geometric fact: if $\mathbb{R}P^n$ can be immersed in $\mathbb{R}^{n+1}$ then for some $k$, $n=2^k-1$ or $n=2^k-2$. My initial ...
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votes
3answers
708 views

Isometries of $\mathbb{S}^n$

How to prove so elementary (elementary = without using the concept of geodesic) that an isometry of $\mathbb{S}^n$ is a restriction on $\mathbb{S}^n$ of an isometry of $\mathbb{R}^{n+1}$ ? EDIT: ...
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0answers
188 views

Extending Tensor Fields defined on Manifolds to Ambient Space

I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me. The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
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1answer
283 views

Proving a curve is a geodesic.

I am really stuck on the following question. Let $ \ \gamma : I \longrightarrow M \ $ be a non-constant (i.e $ \ \gamma'\ $ is not identically zero) geodesic. Show that a reparametrization $\ ...
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1answer
556 views

Precise definition of conformal structure based on a Riemannian metric on a Riemann surface

As I read the literature, I keep having some doubt about what a " conformal structure on a Riemann surface " exactly means. ( You can assume all the Riemann surface in this literature have universal ...
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0answers
184 views

Hopf-Rinow theorem

If $(M,g)$ is a riemannian manifold. $M$ is complete(geodesically) then any two point can be joined by a geodesic. Geodesic($\gamma(t))$ is smooth curve such that ...
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1answer
147 views

Non unique solution for Ricci flow equation

Why completeness is important for the uniqueness of solution to Ricci flow? For example, if $M$ is the open unit disk in $\mathbb{R}^2$ and $g(0)$ is the Euclidean metric, and hence not complete. Why ...
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1answer
512 views

Cigar soliton solution

In wikipedia, it has been written that an important 2-dimensional example of Ricci flow over $M=\mathbb{R}^2$ is given by $g((x,y),t)=\frac{dx^2+dy^2}{e^{4t}+x^2+y^2} \;\;\; (\star) $ Here are my ...
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1answer
438 views

Determining the “positivity” or “negativity” of Chern class (number?) of zero-sets of homogeneous polynomials

If $\Omega$ is the curvature 2-form on a $n-$manifold, then I would think that the Chern classes (forms), $c_k$ are defined as, $det(I + \frac{it\Omega}{2\pi}) = \sum c_k t^k$ I would like to ...
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1answer
311 views

question on connection on tensor bundle induced by a linear connection

I have a question regarding the definition of the covariant derivative of tensor fields, as given by John Lee in the book Riemannian Manifolds: An Introduction to Curvature On page 53 he states the ...
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177 views

Riemannian connections: how to understand $\overline{\triangledown}_X \langle Y,Z \rangle$

I have a question regarding a comment in Lee's book "Riemannian Geometry - an Introduction to Curvature". On page 52, Lee introduces the Euclidean Connection as the map $\overline{\triangledown} : ...
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1answer
263 views

Upper bound on covering multiplicity

Suppose $M^n$ is a manifold with $Ric(M)\geq(n-1)k$ for some $k \in \mathbb{R}$. For a given point $p \in M$, let $B_p(r)$ denote the metric ball of radius $r$. Given $B_p(r)$ and $\varepsilon > ...
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votes
1answer
426 views

How to calculate scalar curvature in a local chart

I want to find a complete manifold with infinite diameter which has uniformly positive scalar curvature. And I want to show that $M^n = S^2(r) \times \mathbb{R}^{n-2}$ with $n \geq 3$ is an example ...
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210 views

Riemannian Connection (Very basic question)

We know that a connection $\nabla$ in a manifold M hashas the purpose of performing the same role as the covariant derivative of vector fields of surfaces in $\mathbb{R}^3$. Such analogies are ...
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2answers
149 views

help on connections

In the book on Riemannian Geometry by John Lee ("Riemannian manifolds: an Introduction to Curvature") the author gives an exercise on page 54 involving connections: Let $\triangledown$ be a linear ...
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1answer
459 views

Need references on Cartan's method of moving frames.

Could anyone suggest a book or a paper containing a good, modern treatment to the Cartan's method of moving frames. Especially, I am interested in its use in studying geometric properties of surfaces ...
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1answer
331 views

The behavior of all unit speed geodesics on a surface of revolution.

In the $xz$-plane of $\mathbb R^3$, consider the closed non-singular curve $\gamma$ which is the image of the function $$t\mapsto (1+2\sin^2(t))(\cos(t),0,\sin(t)).$$ (Note that $\gamma$ is invariant ...