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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Why is $\gamma(t)=(0,t)$ a geodesic in the hyperbolic plane?

I'm having trouble to understand a very simple fact of the book of DoCarmo "Riemannian Geometry". In the page 73 he calculates the geodesics of the hyperbolic plane: $$ \mathbb{R}^+_2 = \{ (x,y) \in \...
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12 views

Reparametrization of a curve in the hyperbolic plane

I'm trying to solve a problem of a course in Riemannian Geometry that I'm taking. I'm considering $$\mathbb{R}_2^+ = \{(x,y) \in \mathbb{R}^2 : y > 0 \}$$ With the metric given by: $$g_{ij} = y^{-...
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45 views

conformal structure of a disc

I wonder if the conformal structure of the unit disc $D^2=\{(x,y):x^2+y^2\leq 1\}$ is unique. More precisely, given a Riemannian metric $g$ on $D^2$, is it always true that $g=e^{2u}g_0$, where $g_0$...
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1answer
54 views

Fundamental group and curvature

Is there any paper about the $\pi_1$ group and curvature ? Because how close a curve depends on the curvature near the curve . I think there must have some condition which decide whether there is ...
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1answer
74 views

Why is $T$ in $\mathcal T_2^1(M)$?

Let $M$ be smooth manifold and $\nabla$ an affine connection on $M$. Then the torsion tensor of $\nabla$ is the map $T:\mathcal T(M)\times\mathcal T(M)\to\mathcal T(M)$ and $$T(X,Y)=\nabla_XY-\...
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21 views

Sublinear functions on a Riemannian manifold

I would like to know if there is any notion of sublinear function or subadditive function for Riemannian manifolds. Thank you!
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114 views

Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
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Existence of solutions of $\Delta u = f$ such that $|u|_{L^\infty} < \infty$ if $|f|_{L^\infty} < \infty$.

I am struggling with figuring out the details of proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor. Setting is as follows. Let $\Omega$ be a noncompact Riemann ...
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1answer
36 views

Compute Christoffel symbol of $\mathbb S^2$.

Let $$(x,y,z)=f(\theta,\gamma )=(\sin \varphi\cos\theta,\sin\varphi\sin\theta,\cos \varphi).$$ Therefore, $$\frac{\partial }{\partial \theta}=(-\sin\varphi\sin\theta,\sin\varphi\cos\theta,0)$$ $$\...
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1answer
23 views

Geodesics on $SO(n)$

I'm trying to prove the following exercise about the ortogonal symmetric group $SO(n)$. I have been able to prove the first two sections of the exercise but I got stuck on the third. I don't ...
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1answer
39 views

Curvature Scalar in Riemannian Space

Suppose that Riklm=a(gilgkm-gimgkl ) on some four dimensional Riemannian space and a is a constant. Question: Show that for the curvature scalar we have R=-12a. What I know from calculating the ...
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1answer
33 views

Question about connections and usual derivative.

Let $\nabla $ a covariant derivative. What does mean "in the normal coordinate, $\nabla $ is equivalent to the usual derivative". I recall that the normal coordinate is coordinate system on a normal ...
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0answers
86 views

Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
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1answer
32 views

Simple proof of the existence of lines in the hyperbolic space

Let $\mathbb{H}^n$ be the hyperbolic space defined as warped product: $$ g_{\mathbb{H}^n} = dr^2 + \sinh(r)^2 g_{\mathbb{S}^{n-1}}. $$ What is the easiest way to show that there exist at least one ...
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Questions about $1-$ form $\beta$ on $(M,\alpha)$ such that satisfies $\nabla\beta(X,Y)=|| \beta^\sharp||^2 \alpha(X,Y)-\beta(X)\beta(Y)$

Suppose Riemanian manifold $(M,\alpha)$ and 1-form $\beta$ such that $$\nabla\beta(X,Y)=|| \beta^\sharp||^2 \alpha(X,Y)-\beta(X)\beta(Y)$$ and Questions are : $1.$ $\beta $ is closed and it has ...
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44 views

Bianchi identity proof : why we can consider $[X,Y]=[Y,Z]=[X,Z]=0$?

I recall that the Riemann curvature tensor is defined by \begin{align*} R:\Gamma(M)\times \Gamma(M)\times \Gamma(M)&\longrightarrow \Gamma(M)\\ (X,Y,Z)&\longmapsto [\nabla _X,\nabla _Y]Z-\...
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Does a tensor quantity only depend on the vector field at a point?

In Riemannian geometry there are many functions that have vector fields as arguments. Some (like the curvature tensor) are tensorial and some (like $<\nabla_{X}Y,Z>$) are not. Does the value of ...
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1answer
44 views

Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
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2answers
75 views

Show that geodesic equation is given by $\ddot x^k +\Gamma_{ij}^k \dot x^i\dot x^j=0$

I know that $\gamma $ is a geodesic if and only if $$\nabla _{\dot \gamma}\dot\gamma =0.$$ Using this, I'm trying to re find the equation $$\ddot x^k +\Gamma_{i\ell}^k \dot x^i\dot x^\ell=0,$$ but I ...
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If $\pi:\widetilde{M}\to M$ is a covering map, $M$ is complete iff $\widetilde{M}$ is complete

Let $\widetilde{M}$ be a covering space of a Riemannian manifold $M$. Show that $\widetilde{M}$ has a Riemannian metric such that the covering map $\pi:\widetilde{M}\to M$ is a local isometry. Then $\...
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Is it always possible to always choose coordinates so that the curvature is locally zero?

I would have thought that this was completely possible as manifolds are so "soft" and the only problems would have been global ones (like Gauss Bonnett etc). But I've never seen the phrase "of course ...
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Killing fields and the Laplacian on $S^{2n + 1}$

If $X_{ij} = (x_i\partial_j - x_j\partial_i|_{S^n}$, where $(x_1,..,x_{n + 1}) \in \mathbb{R}^{n + 1}$, then it is known that the Laplacian $\Delta$ on $S^n$ is given by $\Delta = \sum_{i \neq j} X_{...
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1answer
59 views

Why $\nabla _{\dot \gamma (t)}Y_t=\dot x^i \frac{\mathrm d a^j(t)}{\mathrm d t}\partial _j+\dot x^i a^j\nabla _{\partial _i}\partial _j$

Let $M$ a smooth manifold and $\nabla $ a connexion. Let $\gamma :[a,b]\longrightarrow M$ a $\mathcal C^\infty $ curvature. I recall that if $X,Y\in \Gamma(M)$, and $f,g\in \mathcal C^\infty (M)$, ...
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36 views

Hyperbolic isometries and finite order elements

I'm reading a paper and I'm uncertain about one of its claims. I was wondering if someone could clarify. Namely, it states that for a discrete subgroup of $\text{Isom } H^n$, the finite order elements ...
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1answer
34 views

Pullback of the metric on $\mathbb S^n$ on $\mathbb R^n$.

Let $\varphi:\mathbb R^n\longrightarrow \mathbb S^n$ the inverse of the stereographic projection, i.e. $$\varphi(y)=\left(\frac{2y}{\|y\|^2+1},\frac{\|y\|^2-1}{\|y\|^2+1}\right).$$ What I'm trying to ...
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0answers
22 views

Geometric meaning of a curvature equation

In Do Carmo's differential geometry, he proves a lemma about Riemannian curvature tensor. Let $f: A\in \mathbb R^2 \to M$ be a parametrized surface and let $(s,t)$ be the usual coordinates of $\...
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1answer
50 views

Gauge condition equivalent to condition that coordinate functions satisfy wave equation to first order

Let $\eta_{ab}$ be the metric of special relativity and let $x^\mu$ be global inertial coordinates of $\eta_{ab}$. Let $\gamma_{ab}$ be a small perturbation of $\eta_{ab}$. How do I see that the gauge ...
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A differential equation and a Jacobi field

I have a question about the following answer here: Zero Sectional Curvature implies exp is a local isometry We have $w\in T_v(T_pM)\cong T_pM$, the geodesics $\gamma_s(t)=\exp_p(t(v+sw))$ and the ...
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27 views

Relation between inner product and wedge product

$M$ is an Riemannian n-manifold, $p\in M$. $x, y\in T_pM$. Is it true that $\langle x,x \rangle \langle y, y \rangle-\langle x, y \rangle^2=|x \wedge y|^2$? I'm reading Do Carmo's Riemannian ...
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12 views

Steady Ricci solitons with negative Ricci curvature

I am looking for an example of steady Ricci soliton with negative Ricci curvature. Any help will be appreciated.
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1answer
18 views

Extending Riemannian Manifold to Boundary

If you have a Riemannian manifold $(M,g)$ (maybe with other assumptions as need), is there a natural way to extend it to a smooth manifold with boundary? For example, the Lobachevsky space viewed as ...
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1answer
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Hodge decomposition on Riemann surface

On a compact Riemannian manifold $M$ the Hodge decomposition takes the form $$\Omega^k(M)=d\Omega^{k-1}(M)\oplus\mathcal{H}(M)\oplus d^*\Omega^{k+1}(M)$$ Where $d^*$ is the adjoint of $d$ w.r.t. the ...
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1answer
16 views

Covariant derivative of a tensor field

Let $F$ be a tensor field of type $(0,2)$ on a Riemannian manifold (like a Riemannian metric). Let $\gamma$ be a geodesic on $M$ and let $e(t)$ be a parallel transport along $\gamma$. I want to find a ...
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1answer
28 views

Locally symmetric spaces and the curvature tensor

Let $M$ be a Riemannian manifold. Suppose $\nabla R=0$ where $R$ is the curvature tensor (we then say $M$ is locally symmetric). Then if $\gamma$ is a geodesic of $M$ and $X,Y,Z$ are parallel vector ...
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1answer
22 views

Parallel Transport and Affine Connection

$M$ is is a Riemannian Manifold with an affine connection $\nabla$, $X$ is a vector field of which the restriction on the curve $\gamma$ is parallel. Fix some point $\gamma (t) $ on the curve. Is it ...
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votes
1answer
88 views

On the proof of that fixed point set of an involution is a submanifold

Let $M$ be a smooth manifold, and let $f:M\to M$ be a smooth involution (i.e. $f^2=\text{id}$). If we introduce a Riemannian metric on $M$ so that $f$ is isometry, we can prove easily that the fixed ...
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1answer
52 views

Normal variation of embedded surfaces

Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by $$\phi(p,t)=\exp_p(...
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1answer
20 views

Trace of Ricci flow equation

The Ricci flow equation as is known is given as: $\partial_t g_{ij} = -2R_{ij}$. If I take the trace/contract the indices of both sides, does this imply that: $\partial_t g = -2R$, where $g$ is ...
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How does the existence of a unique geodesic in non-positive curvature follow from Cartan-Hadamard?

Cartan-Hadamard states that a non-positively curved Riemannian manifold is covered by $\mathbb{R}^n$. How does it follow that in each homotopy class of paths from $x$ to $y$ there exists a unique ...
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1answer
45 views

Visual explanation of the indices of the Riemann curvature tensor

I'm trying to understand the meaning of the Riemann curvature tensor, but I don't seem to be ready yet to understand the detailed rigorous definition. Anyway, I managed to understand (this gif was ...
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Volume element induced by metric tensor

I am using Wald's General Relativity. In Appendix B he says that given a smooth Lorentzian manifold $(M, g)$ then there a natural choice of volume element $\epsilon$ specified up to sign by $\epsilon^{...
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Existence of geodesic frame

I have a question about the answer to this question I don't know how to get the second equality in $$\nabla_{E_i} E_j (\gamma(t)) = \nabla_{\gamma'} E_j (\gamma(t)) = 0$$ I tried to use the ...
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20 views

Gauss Theorem - Riemannian Geometry

I want to prove Gauss' Theorem: "Let p $\in$ M and x,y orthonormal vectors of $T_p M$. So k(x,y)-$\bar{k}$(x,y)= < B(x,x),B(y,y)> - |B(x,y)|$^2$. From Riemannian Geometry, Manfredo do Carmo We ...
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1answer
31 views

Induced metric on a one-sheet hyperboloid

I am trying to find the induced metric on a one-sheet hyperboloid. Suppose we use cylindrical coordinates $(r, \theta, z)$ for the ambient space in which the hyperboloid is embedded. The hyperboloid ...
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17 views

Gauss curvature and bisectional curvature

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where X is a Kahler manifold. Then is it correct that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega)$$ where ...
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2answers
36 views

How does Riemannian geometry yield the postulates of Euclidean Geometry?

I am reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt: The fact that geometry can be established analytically and independently of any special ...
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Basic question on second covariant derivative

I am having a question on the wikipedia article https://en.wikipedia.org/wiki/Second_covariant_derivative Using the notation therein I don't get why $(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\...
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1answer
38 views

Demonstration of a basic formula involving differential forms

I'm writing some notes on Lie Groups and I'm not sure if I should demonstrate this formula or not. Assume $\omega$ is a differencial form and $X,Y$ fields con a Manifold M, is there a simple way to ...
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1answer
23 views

the heat equation for mappings between closed Riemannian manifolds

Let $M$ be a closed (smooth) Riemannian manifold. Then we have the following existence and uniqueness theorem for the heat equation on $M$, which is considered more or less a standard result: Let $0&...
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1answer
26 views

Functions with nonnegative laplacian on Rimannian manifold.

I am doing the exercises in Do Carmo's "Riemannian Geometry". I am stuck on exercise 3.12 which states the following: Let $M$ be a compact orientalbe Riemannian manifold which is also connected. Let $...