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 ...

learn more… | top users | synonyms

1
vote
0answers
10 views

Parallelizable open dense subset and integration

In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral ...
1
vote
2answers
47 views

Why the differential of exponential map is the identity.

Let $M$ a manifold and $T_pM$ it's tangent plan at $p$. We defined \begin{align*} \exp_p:U_p\subset \Omega _p&\longrightarrow M\\ V&\longmapsto \gamma _V(1) \end{align*} where $\gamma _V:I_V\...
3
votes
1answer
48 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
1
vote
0answers
9 views

Geodesic flow generated by Riemannian distance function

This is an exercise in AC da Silva's Lectures onn Symplectic Geometry; I am having trouble showing the following. $(X,g)$ is a geodesically complete manifold, and $f: X \times X \to \mathbb{R}$ is ...
-1
votes
0answers
26 views

Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
1
vote
1answer
30 views

how to define complete pseudo-riemannian manifold

In riemannin geometry, we define distance function by minimizing the length of curves. However we have nondefinite metric on psedu-riemannian manifold, so we cannot define a length of a curve as ...
2
votes
0answers
14 views

Flag Curvature in Finsler Geometry

Does anyone know what is the flag curvature in Finsler geometry? I looked for this definition, but I don't find any answer.
1
vote
1answer
24 views

About transformations of the metric: should we use the old or the new one to raise/lower indices?

Let $(M,g)$ be a (Pseudo-)Riemannian manifold. If I perform a transformation on the metric, getting a new metric $\tilde{g}$, which metric should I use to raise and lower indices? As I understand, ...
2
votes
1answer
41 views

Confused About Indices in Deriving Curvature

Asking about a step regarding indices in deriving the Curvature tensor from the geodesic equation. Starting from $$ \frac{d v^a}{du} = - \Gamma^a_{bc}v^b \frac{dx^c}{du}$$ we integrate $$v^a(u) = ...
3
votes
0answers
25 views

existance of gradient conformal vector field on riemannian manifolds

Let ‎$ ‎(M,g)‎ $ ‎be a Riemannian manifold. A gradient conformal vector field on ‎$ ‎M‎ $‎ is a conformal vector field ‎$ ‎X‎ $‎ which is at the same time the gradient of a function on ‎‎$ ‎M‎ $ ‎:‎ \...
0
votes
0answers
8 views

Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ : $$ dist\big(\gamma(s), \gamma(t)\...
0
votes
1answer
37 views

Riemannian connection on Lie groups

Let $G$ be a Lie group with a bi-invariant metric. Then, the Riemannian connection is given by $\nabla_XY=\frac1 2 [X,Y]$ for all $X,Y\in \mathfrak g$. In the proof: Since $\langle X,Y\rangle$ is ...
2
votes
0answers
33 views
+50

Distance function on complete Riemannian manifold.

Let $\left(M, g\right)$ be a complete Riemannian manifold. Let us fix a point $p \in M$ and consider the distance function $$ r(x) := \operatorname{dist}(x, p). $$ I would like to characterize the ...
3
votes
0answers
53 views

Is exponential map locally a diffeomorphism w.r.t. base point?

Let $M$ be a riemannian manifold and $\exp_p: T_pM \rightarrow M$ the exponential map at $p \in M$. At each point $p\in M$, $\exp_p$ can be restricted to a neighborhood $V$ of $0\in T_pM$ so that $\...
2
votes
0answers
37 views

Show that $\operatorname{grad} f(p)=\sum_{i=1}^{n}{(E_{i}(f))E_{i}(p)}$

Let $M$ a Riemannian manifold.Let $X\in\chi(M)$ and $f\in\mathcal{D}(M)$. Define the gradient of $f$ as the vector field $\operatorname{grad} f$ in $M$ define by $$\langle\operatorname{grad} f(p),v\...
0
votes
1answer
22 views

Why the induced metric from the lie algebra of lie group $G$ is left invariant.

we say that a Riemannian metric on $G$ is left invariant if $<u,v>_y = <d(L_x)_y u,d(L_x)_y v>_{L_x(y)}$ to introduce a metric on $G$, take any arbitrary inner product $< , >_e$ on ...
1
vote
0answers
16 views

remannian connection \lamba_XX=0

$M$ is a Riemannian manifold with Riemannian connection $\nabla$. $X$ is a vector field on $M$. It is not in general true that $\nabla _XX=0$, for example a geodesic $\gamma$ satisfies $\nabla_{\frac{...
0
votes
1answer
50 views

Vanish christoffel symbols implies $g_{ij,k}=0$

Let $(M,g)$ be a pseudo-riemann manifold and $(U,\psi=(x^1,\ldots,x^n))$ a local chart around some point $p$ in $M$. It is easy to show that if $\partial g_{ij}/\partial x^k=0$ in $p$ for all $i,j,k$ ...
1
vote
0answers
35 views

Sectional curvature of 2-manifold

This is problem7, chapter 5 of Do Carmo's Riemannian geometry. Let $M$ be a Riemannian two manifold, $p\in M$ , $exp_p$ is a diffeomorphism on a neighbourhood of origin $V\in T_pM$. Let $S_r(0)\...
2
votes
0answers
36 views

What will happen if evolve metric under Ricci flow on general manifold?

Because the scalar curvature under Ricci flow evolve by $$ \partial_t R=\Delta R+ 2|Ric|^2 $$ I treat it as heat equation with heat source . So, no matter the heat source is very hot or not , the ...
0
votes
1answer
20 views

normal connection on immersed hypersurface vanishing

I am studying Riemannian geometry using Do Carmo's book. I am learning about isometric immersions right now, and I got stuck with the following claim about Codazzi's equation. Let $f:M^n \to \...
3
votes
1answer
18 views

Connection after a metric rescaling and compatibility

It's known (see here for example) that after a rescaling of the metric $\tilde{g}=e^{2\omega}g$, we can find a new connection $\tilde\nabla$ associated to the new metric: $ \tilde\nabla _X Y = \nabla ...
2
votes
0answers
31 views

Convergence of Discretized Geodesics?

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto \mathbb{R}^D$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the ...
2
votes
1answer
46 views

Lie bracket and inner product

$X, Y, N$ are vector fields on a riemannian manifold $M$. $\langle X, N \rangle(p)=0 $, $\langle Y, N \rangle(p)=0$ at some point $p$. Show $\langle [X,Y], N \rangle(p)=0$. I want to use $[X,Y]=XY-...
2
votes
2answers
70 views

Problem to conceptualize $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$.

I have some little problem to give a conception to $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$ on manifold (like $\frac{\partial }{\partial x}$ as well). For example, ...
2
votes
1answer
37 views

Curvature of the sphere, how can I get it?

I want to compute the curvature of the sphere. I have the following definition : The curvature is given by $$K_p(T_p\mathbb S^2)=\frac{R(X,Y,Y,X)}{\|X\wedge Y\|^2}$$ where $X,Y$ is a basis of $T_p(\...
2
votes
1answer
54 views

Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
0
votes
0answers
25 views

the map from the horizontal bundle is a submersion or an immersion

Let $\pi \colon M \to B$ be a riemannian submersion and $g$ the metric on $M$. Then we get the vertical subspace $ \mathcal{V}_x = \ker d_x \pi$ and the horizontal subspace $\mathcal{H}_x= \mathcal{V}...
1
vote
2answers
30 views

Show that $\hbox{div}(X)=\sum_{j=1}^{n}\langle\nabla_{E_j}X,E_j\rangle$

Let $M$ a riemannian manifold. Let $X\in\chi(M)$ and $f$ a function $C^{\infty}$ in $M$. Define the divergence of $X$ as a function $div X:M\to\mathbb{R}$ given by $\operatorname{div}X(p)=\{\mbox{...
0
votes
1answer
16 views

Do I have to sum twice the symmetric Cristoffel symbols in the geodesic equation?

When writing the equation of the geodesics of a Riemannian manifold in local coordinates I have (I consider the Einstein's summation convention): $$\frac{d^2x^\mu}{dt^2} + \Gamma^\mu_{\nu \rho} \frac{...
1
vote
0answers
29 views

Mean Value Property for harmonic functions

I would appreciate any insights on this matter; Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...
3
votes
0answers
37 views

Is the space of rigid Riemannian metrics convex?

Let $M$ be a smooth manifold. Let $g_1,g_2$ be two rigid Riemannian metrics on it. (i.e with no isometries except the identity). Is it true that every convex combination of $g_1,g_2$ is also rigid? ...
2
votes
2answers
23 views

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 \...
0
votes
0answers
11 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^{-...
0
votes
1answer
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 $g_0$...
1
vote
1answer
53 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 ...
1
vote
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-\...
2
votes
1answer
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!
7
votes
0answers
113 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 ...
1
vote
0answers
32 views

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 ...
0
votes
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, $$\frac{\partial }{\partial \theta}=(-\sin\varphi\sin\theta,\sin\varphi\cos\theta,0)$$ $$\...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
0answers
85 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 ...
1
vote
1answer
30 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 ...
1
vote
0answers
41 views

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 ...