(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

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668 views

How to find the inverse metrics?

I know one can calculate the inverse of metric tensor $g$ in coordinates as the inverse of it's matrix $g_{ij}$. However what I really liked about differential geometry is how one can actually avoid ...
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1answer
128 views

Product of Riemannian manifolds?

Given two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$ is there a natural way to combine them to be a Riemannian manifold? Some kind of $(M \times N, g^{M \times N})$.
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198 views

How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$. Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$. Prove: Suppose $f\in C^\alpha(M)$ satisfies ...
4
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2answers
262 views

Moving to a conformal metric

Given a generic 2-dimensional metric $$ ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2 $$ what is the change of coordinates that move it into the conformal form $$ ...
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2answers
582 views

Learning differential/Riemannian geometry for PDEs

I know there have been threads on which books to learn DG/RG from but hopefully this is sufficiently different to avoid closure. Can anyone recommend a book to learn DG/RG (whichever is appropriate) ...
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1answer
124 views

The number of geodesics of a complete Riemann manifold with non-positive sectional curvature

There is a theorem of Cartan which states that if $M$ is a simply connected, complete Riemann manifold, and that the sectional curvature is everywhere $\leq 0$, then any two points of M are joined by ...
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2answers
199 views

Upper half plane is complete with the Lobatchevski metric

How do I show that the Upper half plane is complete with the Lobatchevski metric? I tried to use the fact that $M$ is complete iff the lengh of any divegert curve is unbounded,but did not get any ...
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1answer
460 views

Question in do Carmo's book Riemannian geometry

This is a question on Do Carmo's book "Riemannian Geometry" (question 7 from chapter 7): Let $f:M\to \bar{M}$ be a diffeomorphism beetwen two riemannian manifolds. Suppose $\bar{M}$ complete and ...
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1answer
147 views

Question about an isometric immersion

This is the question: Let $M,N$ be Riemannian manifolds, such that the inclusion $i:M \to N$ is a isometric immersion. Give a example where the inequality $d_M > d_N$ may occur. I thought ...
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0answers
85 views

Manifold contains a totally geodesic closed hypersurface

Let $(M^n,g)$ be a closed simply-connected positively curved manifold. Show that if $M$ contains a totally geodesic closed hypersurface (i.e., the second fndamental form or shape operator is zero), ...
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0answers
83 views

uniqueness of asymptote in manifold

Question 1 Let $M$ be a complete, noncompact Riemannian manifold, a ray $\gamma:[0,\infty) \rightarrow M$ starting from $p$, and a point $x \in M$ such that the asymptote $\widetilde{\gamma}$ ...
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1answer
68 views

Differentiable map conserving geodetic lines which is no isometry

I am looking for a differentiable map $f: S^n\rightarrow S^n$, which conserves the geodetic lines of the standard metric on $S^n$, but is no isometry. The geodetic lines on $S^n$ should be the great ...
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1answer
277 views

How long will it take me learn and understand differential (Riemannian) geometry for PDEs? [closed]

I want to learn DG and RG so I can use them in PDEs. Atm I have no knowledge of either DG or RG (and not that much of PDEs either..) but I have a couple of books (John M Lee and Loring). If I spend ...
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1answer
136 views

Do elliptic operators on Riemannian manifolds have a regularizing effect?

I'm working on my master thesis and need to handle some spectral theory of the Laplace operator on compact Riemannian manifolds and especially on the sphere. While investigating essential ...
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1answer
332 views

Is it true that the Laplace-Beltrami operator on the sphere has compact resolvents?

We consider the Riemannian structure on the sphere $\mathbb{S}^n$ seen as a submanifold of $\mathbb{R}^{n+1}$ and the Laplace-Beltrami operator defined on $C^\infty(\mathbb{S}^n)$ by the equation ...
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1answer
127 views

Differential operators on the sphere

The sphere $\mathbb{S}^2$ is a Riemannian submanifold of the Euclidean space $\mathbb{R}^3$ and as such comes equipped with an array of differential operators, particularly gradient, divergence and ...
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0answers
216 views

Does the curvature determine the metric for all surfaces

In order to not make things even more confusing than they are, I split my two-in-one question into two parts. Here's the second part (the first part is here): Here I asked the question whether the ...
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2answers
279 views

Does the curvature determine the metric?

Here I asked the question whether the curvature deterined the metric. Since I am unfortunately completely new to Riemannian geometry, I wanted to ask, if somebody could give and explain a concrete ...
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1answer
346 views

Second Bianchi identity

This is q. 7 of ch. 4 from Do Carmo's book on Riemannian Geometry . Prove that: $$ \nabla R(X,Y,Z,W,T) + \nabla R(X,Y,W,T,Z) + \nabla R(X,Y,T,Z,W)=0.$$ Let $\{e_i\}$ a geodesic frame on $p$ , it is ...
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3answers
833 views

Expression of the Hyperbolic Distance in the Upper Half Plane

While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia. ...
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1answer
128 views

Things related to the Preissman Theorem

I'm reading the proof of the Preissman Theorem, in Do Carmo's book of Riemannian Geometry. A crucial step in this demonstration is the following lema, Lema: Let $M$ be a compact riemannian ...
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1answer
66 views

Commutator map and the derived series

Let be $G$ a solvable group, let $$ G=G_0\supset G_1\supset\cdots\supset G_k=1$$ be the derived series for $G.$ Is clear that $G_ {k-1}$ is abelian. Now take $b\in G_{k-1}$ e $a\in G_{k-2}$ my ...
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2answers
235 views

Cartan Theorem.

Cartan Theorem: Let $M$ be a compact riemannian manifold. Let $\pi_1(M)$ be the set of all the classes of free homotopy of $M.$ Then in each non trival class there is a closed geodesic. (i.e a closed ...
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1answer
112 views

Rotation of a catenary in $\mathbb{R}^5$

If you rotate a catenary in $\mathbb{R}^3$, then you get a catenoid. To show: If you rotate the same catenary in $\mathbb{R}^5$, then you get a 4-dimensional hypersurface. I'm not sure, if I got this ...
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2answers
301 views

Riemannian metric making a given function harmonic

I have a nice 3-manifold (closed, oriented) which fibers over the circle, i.e. we are given a fibration $f:M\to S^1$. Apparently $M$ should admit a metric such that $f$ is harmonic. I don't quite ...
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0answers
55 views

Relation with Jacobi fields in a small neighbourhood of some point in a complete manifold

I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
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2answers
313 views

Explicit expression for eigenpairs of Laplace-Beltrami operator

In $R^n$, the Laplace-Beltrami operator is just the Laplacian, and its eigenstructure is well known. There are also explicit expressions for the eigenvalues/eigenvectors of the Laplace-Beltrami ...
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1answer
105 views

Local equation for Jacobi fields

Let $\gamma:[0,1] \longrightarrow M$ a geodesic in M, and V a Jacobi field. Let $(E_1(t),...,E_n(t))$ an orthonormal base of $T_{\gamma(t)}M$ with $E_1(0)=\frac{\gamma'(0)}{||\gamma'(0)||}$. Define ...
3
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1answer
373 views

How to induce a connection on a submanifold?

Suppose an affine connection is given on a smooth manifold $M$ and let $N\subset M$ be an embedded submanifold. Is there a canonical way of defining an induced connection on $N$? In classical ...
2
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1answer
141 views

Is every closed convex subset a sublevel set?

Is it true that on every Riemannian manifold $M$ (whether compact or merely complete), every closed convex set C in M is the sublevel set $f((-\infty,t])$ of some convex function $f : M \rightarrow ...
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1answer
118 views

A question about covariant derivative of a tensor

Let $R'$ be a tensor of order 4 in a riemannian manifold $M$ defined by: $R'(W,Z,X,Y)=\langle W,X \rangle \langle Z,Y\rangle - \langle Z,X\rangle \langle W,Y\rangle $ And let $R$ be the curvature ...
3
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2answers
158 views

Relation between uniform convergence of curves in a manifold and homotopy

I was working through some things in riemannian geometry and I had this doubt: Let $M$ be a closed riemannian manifold, $H$ an embedded submanifold and $V$ be its $\varepsilon$-tubular neighborhood. ...
3
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0answers
148 views

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
115 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
82 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
248 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
185 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
79 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
187 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
629 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
190 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
445 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
69 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
146 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
122 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 ...
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2answers
354 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
151 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
50 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
85 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
125 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: ...