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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596 views

parallel transport preserves orientation

In my text its written that parallel transport on a Riemannian manifold preserves orientation. Can someone clarify what does that mean? I am confused about this notion.
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1answer
179 views

condition for compatible connection on a Riemannian manifold

Prove that connection $\nabla $ on a Riemannian manifold $M$ is compatible with metric iff $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ for every smooth vector fields $X,Y,Z$. I am confused about ...
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290 views

Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature

Let's suppose we have a Riemannian $n$-manifold $(N,g)$ and an immersed surface $f:\Sigma\rightarrow N$, with genus zero, equipped with the induced metric. Let's further assume that the ambient space ...
2
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1answer
45 views

What are $\partial/\partial f^j$ in Jost's definition of the differential mapping?

Let $M$ be a $d$-manifold and $x_0=(x^1,x^2,\cdots, x^d)\in M$, Jost defines the tangent space at $x_0$ to be \begin{equation}\{x_0\}\times \operatorname{span}\left\{\frac{\partial}{\partial ...
2
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1answer
194 views

The smoothness of distance function

$(M,g)$ is a Riemannian manifold and $N$ is a submanifold of $M$, then is the function $r(x)=\mathop {\min }\limits_{y \in N} d(x,y)$ smooth near $N$? ($d(x,y)$ is the distance function induced by ...
3
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249 views

The Riemannian metric on manifold with boundary

$(M,g)$ is a Riemannian manifold with non-empty boundary and $DM$ is the double of $M$, is there a Riemannian metric $G$ on $DM$ such that $g=i^*G$? ($i$ is the inclusion from $M$ to $DM$)
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1answer
145 views

Necessary and sufficient condition for a vector field to be a coordinate vector field

Let $X$ be a vector field on a manifold $M$. Is there a necessary and sufficient condition on $X$ for it to be locally equal to the coordinate vector $\partial_j$ for some coordinate system? For any ...
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0answers
73 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
2
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1answer
291 views

Weitzenböck Identities

The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
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177 views

What are spinor fields?

For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as $$ T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}} $$ with local coordinate ...
3
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1answer
171 views

Volume of a Riemannian manifold and its relation to the area

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem): If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...
2
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1answer
57 views

Question on tensor calculation in Reimannian geometry

Given a Riemannian manfiold $M$ with metric $g=(g_{i,j})$. Let $T=T_{A,B,C,\dots}^{a,b,c,\dots}$ be a tensor on $M$. I would like to compute for example $T_{A,B,C,\dots}^{a,n,c,\dots}g_{n,m}$. We ...
2
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1answer
45 views

Metric is locally constant

If $(M, g)$ be a Riemannian manifold. My doubt is the following: For each $p\in M$, there is a coordinate chart $(U, (x_1,x_2,.., x_n))$ such that $g = \sum dx_i\otimes dx_j $
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votes
1answer
64 views

Continuity of the lengths of geodesics on a closed manifold

Let $M$ be a compact Riemannian manifold and let $SM$ be its sphere bundle, $$SM = \{(x,\xi) \in TM : \|\xi\| = 1\}.$$ There is a well-defined function $\ell : SM \rightarrow [0,\infty)$ defined by ...
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2answers
340 views

Explain the Stokes -theorem from differential from into Integral form

I want to understand the Stokes -theorems deeper. I am trying to understand the operation from $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ to ...
5
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2answers
591 views

orthonormal vector fields

In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$? It seems ...
6
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1answer
260 views

Parabolic elements correspond to punctures

In Mapping Class Group by Farb and Margalit page 22, they say: Let $S$ be a hyperbolic surface. If a non-trivial element of $\pi_1(S)$ is represented by a loop (up to homotopy) around a puncture, ...
4
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2answers
323 views

Is there such a thing as discrete Riemannian geometry?

General relativity expresses gravity as a curvature in space time, created by the stress energy tensor. $$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu}$$ Given I put the fact that energy is ...
5
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3answers
531 views

Geodesics of a “diagonal” metric

Are there any relations that exist to simplify Christoffel symbols/connection coefficients for a diagonal metric which has the same function of the coordinates at each entry? In other words, I have a ...
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1answer
178 views

Given lattice G; parameters of torus R^2/G?

This should be a simple, known result, but I can't seem to find it. Given a lattice $\Gamma = m\mathbb{Z} \times n\mathbb{Z}$, $\mathbb{R}^2/\Gamma$ is topologically a torus. For suitable $m$ and ...
6
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1answer
299 views

How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?

I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$ g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ...
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0answers
67 views

Inner product of $p$-forms [duplicate]

Possible Duplicate: Extension of Riemannian Metric to Higher Forms I have no problems with understanding the inner product of 1-forms on a Riemannian manifold. We have a metric tensor, it's ...
2
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1answer
559 views

Proof of Gauss's Lemma (Riemannian Geometry version)

I was self-learning Do Carmo's Riemannian Geometry, there is a step in the proof of Gauss's Lemma what I can't quite figure out. Since $d\,\exp_p$ is linear and, by the definition of $\exp_p$, ...
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2answers
333 views

Does a diffeomorphism between manifolds induce an isomorphism of Sobolev spaces?

Let $M$ be a Riemannian manifold, and define the Sobolev spaces $H^k(M)$ to be the set of distributions $f$ on $M$ such that $Pf \in L^2(M)$ for all differential operators $P$ on $M$ of order less ...
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1answer
151 views

Is Legendre transform related to finding the inverse metrics?

In classical mechanics kinetic part of Hamiltonian is the Legendre transform of kinetic part of Lagrangian. On the other hand kinetic part of Lagrangian is a metric on the configuration space. At ...
5
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1answer
355 views

Harmonic coordinates for Ricci flow

It is customary to use DeTurck's argument (or Hamilton's original one involving the Nash-Moser iteration) for proving local existence of the Ricci flow. I am wondering why one cannot use harmonic ...
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1answer
1k 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 ...
2
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1answer
160 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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2answers
252 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
376 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
767 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
167 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 ...
2
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3answers
257 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 ...
5
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1answer
633 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
160 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
88 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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90 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 $\widetilde{\gamma}$ ...
2
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1answer
75 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 ...
1
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1answer
321 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 ...
6
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1answer
148 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 ...
5
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1answer
382 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
145 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 ...
3
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0answers
266 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 ...
5
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2answers
364 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 ...
3
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1answer
440 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 ...
5
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3answers
1k 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. ...
5
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1answer
161 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
67 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 ...
8
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2answers
252 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
132 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 ...