Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

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27 views

Taylor expansion on a Manifold

Is there a way to define the Taylor expansion of a function $f:\mathcal{M}\rightarrow\mathbb{R}$, where $\mathcal{M}$ is a smooth manifold? I'm looking for a free coordinate definition. I guess it is ...
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1answer
34 views

What is $dudv$ in the metric tensor?

In the definition of a $m$-dimensional Riemannian manifold $(M,G)$, if $(U;u^i)$ is a local coordinate system of $M$, the tensor field $G$ on $U$ can be written as $$ G = g_{ij}du^i\otimes du^j\;\tag{...
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1answer
37 views

Geodesic equation without metricity

If we relax the metricity condition, what change will be seen in the geodesic equation ? For instance, when the torsionless connection is relaxed while keeping the metricity intact, we see that the ...
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18 views

Isomorphism of the space of mixed (k, l+1) - tensors

Let $V$ be a finite-dimensional vector space. There is a natural isomorphism between $T^{k}_{l+1}(V)$ and the space of multilinear maps $$V^{*l}\times V^{k} \rightarrow V$$ I have found an ...
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8 views

Wave map on manifolds

Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional orientable smooth manifold. Now in a paper ...
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1answer
25 views

Computation of the second derivative of the Jacobian of the change of the coordinates

I am trying to understand how to derive the equality $1.135$ of the book "A course in minimal surfaces" by Colding and Minicozzi. I derive $$(g^{ij}g_{ij}')' = (g^{ij})' g_{ij}' + g^{ij} g_{ij}''$$ ...
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10 views

When a Riemannian manifold is geodesically complete

Given the Lobachevsky plane, show that it´s geodesically complete. We have to prove that every geodesic $$γ(t)$$ is defined for all values of t. for exemple, if we have this geodesic $$(b,1+t)$$, ...
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13 views

The associated structure functions - Riemannian Geometry

I was reading a book on Riemannian geometry and in the section where he talks about Cartan's structural equations I came across the following definition: Let $\{X_1,\ldots, X_n \}$ be a field of ...
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1answer
14 views

Action torsion elements in the fundamental group of geometric orbifolds

In chapter 2 of Three-dimensional Orbifolds and Cone-Manifolds, theorem 2.26 states that complete geometric orbifolds $Q$ modeled on $(G,X)$, whith $X$ simply connected, are such that the holonomy ...
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23 views

Meaning of Lie Bracket of vector fields along curves

I'm reading about Riemannian cubic splines, and the derivation of the Euler-Lagrange equations describing these curves from variational principles. We let $\alpha: [0, T] \times (-\epsilon, \epsilon) \...
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1answer
24 views

If $V^{\mu}$ is a killing vector, then is $∇_μ V^{\mu} = 0$?

Working with the Levi-Civita Connection and a symmetric metric I want to show that if $V^{\mu}$ is a killing vector, then $∇_μ V^{\mu} = 0$. I am not sure how to show this fact, but I believe it to ...
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1answer
54 views

Curvature of Lobatchevski space

Consider the Lobachevsky plane defined as $$\mathbb{R}^2_+ = \left\{{(x,y)\in{\mathbb{R}^2};y>0}\right\}$$ with the metric given by $$g_{11} = g_{22} = \frac{1}{y^2}, g_{12} = 0$$ Calculate the ...
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18 views

Is there a definition of limiting flatness on a manifold?

I am not talking about local flatness. Local flatness means an entire chart can be mapped to a flat euclidean space. For this reason for example a sphere is not flat, because any chart will still be ...
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24 views

How to find geodesics by using of an isometry?

Let $P:(\mathbb{R}^3,\mathcal{D})\to(\mathbb{R}^3,h)$, where at each point $p=(x,y,z)$, $$P(p)= \left(\begin{matrix} 1 & 0 & 0\\ 0 & \cos \alpha(p) & -\sin \...
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1answer
22 views

term for distance preserving up to scale

Is there a common/established term for distance preserving up to scale? E.g. Consider two Riemannian manifolds $M \subset \mathbb{R}^m$ and $N \subset \mathbb{R}^n$ equipped with metrics $g_M$ and $...
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26 views

Lipschitz maps cannot increase the volume of a Borel set by a factor greater than $k^n$.

Let $(M,\,g,\,d,\,vol_g)$ be a Riemannian manifold with metric $g$, geodesic distance $d$ and volume form measure $vol_g = \sqrt{\det(g_{ij})}\cdot m$ ($m$ = Lebesgue measure) and a Lipschitz map $f:K\...
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8 views

Simplifying the Dirac Equation with a Perturbed Metric

I think I heard it mentioned that one could simplify the Dirac equation by taking the metric to be the perturbation of some simple metric (for example, a perturbation of the Schwarzchild metric): $g_{...
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29 views

Robertson Walker metric, covariant derivative

I am currently trying to understand the paper "Global Wave Maps on Robertson–Walker Spacetimes" by YVONNE CHOQUET-BRUHAT and don't understand the following part where this $D$ is defined. The context ...
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1answer
23 views

If $S$ is the $(1,1)$ version of the Hessian of $f$, then $L_{\nabla f} S = \nabla_{\nabla f} S$

Let $(M,g)$ be a Riemannian manifold and let $\nabla$ be the Levi-Civita connection on $M$. Let $f: M \to \mathbb{R}$ be a smooth function and let $$S(X) = \nabla_{X} \nabla f $$ be the $(1,1)-$tensor ...
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24 views

Property of Parallel transport in Manifold

Let M be an Hadamard Manifolds, $C$ a nonempty, closed geodesic convex subset of $M$, $T_xM$ the tangent space of $M$ at $x \in M$ and $TM$ the tangent bundle of $M$. Let $F : C \to TM$ be a smooth ...
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58 views

Does 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions?

I want to know Does 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions? My idea is that one can consider 2-dimensional embedded sub-...
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18 views

Same normal coordinates for two different Riemannian metrics

Let $(M,g_1)$ be a Riemannian Manifold and consider on a normal neighborhood $U$ of $p\in M$ normal coordinates ${x^i}$. I was wondering if it may happens, ${x^i}$ being normal coordinates for some ...
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1answer
50 views

Proving $R_{ij;m}=g^{kl} R_{ikjl;m}$.

In the coordinate $\{x^i\}$, the Riemann curvature tensor can be written as $$ R=R_{ijkl}\,dx^i\otimes dx^j\otimes dx^k \otimes dx^l $$ and the Ricci curvature can be written as $$\text{Ric}=R_{...
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1answer
109 views

Examples of non-positively Curvature Riemannian Manifolds

When I read about complete, simply connected, and connected Riemannian manifolds of non-positive curvature I only find explicit examples of hyperbolic $n$-space and Euclidean space. What are other ...
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1answer
30 views

Riemannian Exponential Map is a Homeomorphism outside the Cut Locus

Let $M$ be a connected and complete Riemannian manifold. The Hopf-Rinow Theorem guarantees that $Exp_p$, for any $p \in M$, is defined on all of $T_p(M)$. Further, this map is a diffeomorphism on a ...
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1answer
50 views

Regarding equivalent definitions of Euclidean Submanifolds in Gallot, Hulin and Lafontaine's book Riemannian Geometry

In the book Riemannian Geometry by Gallot, Hulin and Lafontaine, a proposition which characterises equivalent definitions of submanifolds is given as follows: 1.3 Proposition The following are ...
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19 views

Levi-Civita connection from idempotents

Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
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33 views

How to calculate the Riemannian gradient in practice (optimization)?

In a couple of papers, I saw steepest descent defined by $\mathbf{v} = -\mathbf{G}_p^{-1}\nabla f(\mathbf{p}) \in T_x\mathbb{R}^n$ where $\mathbf{G}$ is the Riemannian metric tensor of an manifold. ...
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1answer
69 views

Argument in a proof for scalar maximum principle

I'm trying to understand how an assertion made in the proof of the scalar maximum principle follows from the compactness of the manifold we're working with. The situation is as follows: I ...
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1answer
33 views

Diameter of triangle in the hyperbolic plane

The diameter of a set $S$ in a metric space $(M,d)$ is defined to be {$\sup d(x,y)|x,y\in S$}. In Euclidean space the diameter of a triangle is the length of the largest side. In the hyperbolic ...
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14 views

The same generalized Gauss map

Let $f:M^{n}\rightarrow \mathbb{R}^{m}$ an isometric immersion of a simply connected Riemannian manifold such that $\Phi\in \Gamma(\mbox{End}(TM))$ is a Codazzi tensor on $M^{n}$, that is, $$(\nabla_{...
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28 views

Transformation rule for Levi-Civita connection on differential forms under conformal metric change

It is well-known, and shown e.g. here or here, how the Levi-Civita connection behaves under a conformal metric change acting on vector fields. What I want to deduce is the representation for ...
2
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1answer
70 views

Second derivative test (and sign of laplacian at critical points) for manifolds

I'm trying to understand in more detail some of the justifications for a proof of the second derivative test for Riemannian manifolds, given below: I've never seen the Laplacian interpreted as an ...
3
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1answer
138 views

Set of isometries of Riemannian manifold is a topological group

I need to show that the set of isometries of a connected Riemannian manifold is a topological group. My work so far has been: Part A: If an isometry $f$ leaves fixed (n + 1) points so close ...
2
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1answer
17 views

Holomorphic functions on Riemann surfaces with boundary

Suppose that $\Sigma$ is a compact Riemann surface with boundary and that $f: \Sigma \rightarrow \mathbb{C}$ is holomorphic*. If $f$ is real-valued along $\partial \Sigma$, is it necessarily true that ...
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0answers
22 views

Non-trivial $\Bbb T^k$ action and generalized Killing vector fields

By this answer I know that The existence of a nontrivial Killing vector field $\xi$ on a compact Riemannian manifold $M$ is equivalent to the existence of a nontrivial $\Bbb S^1$-action on $M$. ...
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0answers
20 views

Naturality for the exponential map for pseudo-Riemannian manifold

Here $(M,g)$, $(\tilde M,\tilde g)$ consists of a smooth manifold $M$ (resp. $\tilde M$) with Riemannian metric $g$ (resp. $\tilde g$). Does the result above hold for pseudo-Riemannian manifolds as ...
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41 views

Riemannian Geometry - Cartan structure equations

I am trying to show that (I'm assuming that the manifold has a Levi connection -Civita $\nabla$): Prove that if the forms $\omega^i$ in a field of orthonormal coframes satisfy $d\omega^i = \alpha \...
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53 views

A confusion about flows

First of all I apologize for asking this basic question. Let $M$ be a smooth manifold with diffeomorphism group $Diff(M)$. My understanding of a (smooth) flow on $M$ is a group homomorphism $\mathbb{R}...
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0answers
60 views

Area convergence

Let $(M^3,g)$ be a compact, connected and oriented Riemannian manifold with boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of free boundary minimal surfaces embedded in $M$ that converges ...
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1answer
38 views

On non-canonical Euclidean metric on $\mathbb{R}^n$

The canonical Euclidean metric on $\mathbb{R}^n$ is as $\sum_{i=1}^ndx_i^2$. We also know that every Riemannian space of zero sectional curvature is Euclidean. Therefore, for example, $\sum_{i=1}^...
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0answers
75 views

Lemma 4.14 Heat Kernels and Dirac Operators

I am trying to work out Lemma 4.14 of the book "Heat Kernels and Dirac Operators" by Berline and Getzler. I am stuck with the proof. For the sake of brevity I am uploading a picture from the book ...
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47 views

Euler equation in a Riemannian manifold

I am trying to prove that the Euler equation for a Riemannian 3-manifold $M$ $$ \frac{\partial u}{\partial t} + \nabla_{u} u = - \operatorname{grad} p \qquad \text (1) $$ is equivalent to $$ \frac{\...
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1answer
31 views

Parallel translation

I read that for the Lorentz metric defined as $d(a,b,c,d)=-a^2+b^2+c^2+d^2$ in $R^4$ the parallel translation (corresponding to the Levi-Civita Connection of $d$) agrees with the parallel ...
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0answers
30 views

Commutation relation between covariant and Lie derivatives

I am currently working on extrinsic riemannian geometry and I am looking for a sort of commutation relation between the covariant and Lie derivatives. To be more precise : considering an hypersurface ...
2
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1answer
48 views

Killing vector fields are affine

Let $(M, g)$ be a Riemannian manifold and let $X$ be a smooth vector field on $M$. We say that $X$ is affine if $L_X \nabla = 0$, where $\nabla$ is the Riemannian connection on $M$. How do we prove ...
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0answers
41 views

calculating geodesics!

I have a Riemannian metric $h$ on $\mathbb{R}$, where $h$ is as $$ \begin{align*} h= \left(\begin{matrix} \frac{1}{\frac12+y^2} & 0 & 0\\ 0 & \frac{1}{...
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46 views

Relation between the differential of a vector field and the covariant derivative

Let $\mathcal{M}$ be a Riemannian manifold and let $X:\mathcal{M} \rightarrow \mathcal{T}\mathcal{M}$ be a smooth vector field, i.e., $X(p) \in \mathcal{T}_p\mathcal{M}$ $\forall p \in \mathcal{M}$. ...
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0answers
43 views

The relation between the geodesics of two Riemannian metrics

I have a Riemannian metric on $R^3$ whose matrix is written as $h=P^TDP$, where $P\in SO(3)$ and $D$ is a diagonal matrix with positive valued smooth functions on the diagonal. Here $SO(3)$ is the ...
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10 views

Critical points of harmonic functions

Let $(M^3,g)$ be a compact Riemannian manifold with boundary $\partial M \neq \emptyset$, and consider a non constant harmonic function $f : M \to \mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$ satisfying ...

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