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

0
votes
2answers
55 views

Laplacian of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how I can write $\Delta^M f$ in terms of $\Delta^{S}f$ ? ((i.e the relation between ...
2
votes
1answer
55 views

Laplacian of a submanifold in an Euclidean space

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$ ($n<m$). Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. ...
0
votes
0answers
43 views

give a metric on $M$ which in not compatible with given connection?

suppose a connection is given on manifold. question is define a metric on $M$ which in not compatible with given connection ? I have now idea how to define such metric . this Problem is related to ...
1
vote
1answer
60 views

Given a $1$-form $\omega$ on $\Bbb R^n$, is there a connection whose torsion is $T(X,Y)=\omega(X)Y-\omega(Y)X$?

Consider $(R^n, g_0 )$, where $g_0$ is the Euclidean metric, and a differential $1$-form $\omega$ on $R^n$. Can this differential form define a connection on $M=R^n$ such that its torsion is $$T(X,Y)=...
1
vote
0answers
21 views

Mean curvature submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap $$ G=S^{N-1}\cap\{x_N>0\} $$ with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...
1
vote
0answers
20 views

Physical Object to Pseudo-Riemannian Manifold

It is well known that Lorentzian mainfold is studied in general relativity. So this raises my curiosity about How about the classical mechanics? Does it correspond to the manifold $\mathbb{R}\times ...
0
votes
0answers
13 views

Propagation of a volume element along a (pseudo) Riemannian metric?

I am considering the propagation of a volume element $\delta V$ along a (probably pseudo)Riemannian manifold. For example, consider the volume element at $\delta V(x_{0}^{\mu})$ . Utilizing the ...
0
votes
1answer
28 views

What is the tangent space o SO(n) [closed]

It should be the kernel of the map $H\mapsto A^TH+H^TA$ at some $A$ such that $A^TA=I$ . But I cant find this Kernel can someone help me?
2
votes
1answer
31 views

Example of a degenerate metric which doesn't have the Levi-Civita connection

The proof of existence of the Levi-Civita connection for pseudo-Riemannian manifolds uses heavily the fact that the metric is non-degenerate - so that $\nabla_XY$ is characterized by all the values $\...
0
votes
1answer
33 views

Nilpotent Lie subalgebra of Lie algebra of Killing vector fields

Suppose $M$ is a smooth manifold with Riemannian metric $g$. Recently I have dealt with some problem which lead me to the following question: Can a Lie algebra of Killing vector fields on $M$ has a ...
2
votes
0answers
25 views

Simplification of Levi-Civita in an orthonormal frame

I have been struggling to understand how picking an orthonormal frame for the tangent space of a Riemann surface with local coordinates ${x_1,x_2}$ simplifies the matrix of one forms associated to its ...
0
votes
1answer
31 views

push forward of the levi civita connection

Let $M$, $M'$ be riemann manifolds with levi-civita connection $\nabla$,$\nabla'$. If $\phi$ is an isometry (global so diffeomorphism too) I want to show: $ \nabla'_{X'} Y'=D\phi (\nabla_X Y) $ where ...
1
vote
0answers
33 views

geodesic flow is proper action

Good evening to everyone. I'm having a problem in the following setting: If I'm having a homogeneous manifold $M=G/K$, where $K \subset G$ is a closed subgroup, I can always find a $G$-invariant ...
1
vote
2answers
57 views

Proving Euler's spiral is an isometric embedding with bounded image

$\newcommand{\al}{\alpha}$ I am trying to prove Euler's spiral is an isometric embedding of $\mathbb{R}$ into $\mathbb{R}^2$ with bounded image. Here is the definition of the spiral: $(*) \, \,\al(t)...
1
vote
1answer
50 views

About the parallel transport

Definition 1: Let $M$ be a differentiable manifold with an affine connection $\nabla$. A vector field along a curve $c:I\to V$ is called parallel when $\dfrac{DV}{dt}=0$ for every $t\in I$. ...
5
votes
1answer
60 views

Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
3
votes
1answer
191 views

How can we define $\partial x_{i_r}^p(X_p^r)$?

Let $M$ be a smooth manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P\subseteq M$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ ...
2
votes
0answers
35 views

Lagrangian densities, Lie Groups and Lie Algebras

I'm quite new to Physics and I was having a look for the first time to the Standard model. I'm not sure if the mechanism that I'm describing is directly from Weyl or from others but what I found quite ...
7
votes
2answers
127 views

Can every Riemannian manifold be embedded in a sphere?

The famous Nash embedding theorem asserts that every closed Riemannian manifold can be isometrically embedded in Euclidean space $\mathbb{R}^n$ for $n$ sufficiently large. Is it true that we can ...
3
votes
0answers
60 views

Two questions about Li-Yau-Hamilton estimate

Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I can't see $Q\ge 0$ when $t=0$. Besides, how to ...
0
votes
0answers
12 views

$S_{2}(f)=0$, with $f$ nonconstant. Applications of the Hessian operator.

Study article R. C. Reilly is entitled Applications of Hessian operator in the Riemann manifold had a doubt in the remark, shortly after the theorem 2 of that Article. The theorem is stated as: ...
0
votes
0answers
15 views

Are the coordinates of geodesic curves in an analytic manifold, analytic functions?

I wonder if the coordinates of geodesic curves in an analytic manifold, analytic functions? Thanks in advance.
1
vote
1answer
35 views

Is a geodesic always a rectifiable curve?

I am not an expert in differential geometry, but I need to know the following If any geodesic that joins two points in a compact and Riemannian manifold is necessary a rectifiable curve, or there ...
0
votes
0answers
23 views

Laplace operator on a compact riemannian manifold $(M^2,g)$ [duplicate]

I'm studying some things about conformally covariant operators and I found this equation that there is an extensive literature about it, second the author. Let be $\Delta_{g_w}$ the Laplace operator ...
4
votes
1answer
53 views

Showing that $\mathbb{R}$ is locally isometric to $S^1$

Show that $f:\mathbb{R}\to S^1$ given by $f(t)=e^{i t}$ is a local isometry between Riemanninan manifolds. So, basically we need to show that for each $p\in\mathbb{R}$ there exists $U\subseteq\...
0
votes
1answer
34 views

Why is the 'line-element' non-integrable?

I'm reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt: A second landmark is the geometry of Riemann, which grew out of the ingenious ...
0
votes
1answer
19 views

Invariance of the rank of the trace of Riemannian curvature under a change of frame

Let $$ R=\begin{pmatrix} R_{11} & ... & R_{1n} \\ &...\\ R_{n1} &...& R_{nn} \end{pmatrix}, $$ where $R_{ikjl}$ is curvature tensor of a Riemannian manifold $(M,g)$ and $R_{ij}=...
1
vote
1answer
72 views

Gradient and Divergence in Riemannian Manifold

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 $divX(p)=\mbox{trace of the linear ...
0
votes
1answer
22 views

Weierstrass transform on the Riemannian manifold

I've read on this Wikipedia article that Weierstrass transform (WT) can be defined on any Riemannian manifold $(M,g)$, but it seems a bit complicated to me. I'm not sure but I guess one can write the ...
0
votes
0answers
35 views

A Riemannian metric on the torus $T^n$

This exercise is from Do Carmo, Riemannian Geometry. Introduce a Riemannian metric on the torus $T^n$ in such a way that the natural projection $\pi:\mathbb{R}^n\to T^n$ given by $$\pi(x_1,...,x_n)...
1
vote
1answer
43 views

Diffeomorphism maps geodesics to geodesics

Let $f:M \to N$ a diffeomorphism between riemannian manifolds of the same dimension. What are sufficient conditions for $f$ to map geodesics to geodesics? Of course, if $f$ is an isometry this occurs, ...
2
votes
2answers
49 views

Formula of a two form for a parallelizable manifold

Let $M^n$ be a parallelizable manifold with the nowhere dependent vector fields $X_1,\ldots, X_n$ forming a basis for the tangent space at each point of $M$. The Lie brackets of these fields are ...
2
votes
0answers
31 views

Newton-Raphson method on manifolds

Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the Newton-...
-2
votes
1answer
29 views

Calculate the Euler-Poincaré characteristic of followin surfaces.

Calculate the Euler-Poincaré characteristic of: An ellipsoid. The surfase $S=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3}:x^{2}+y^{10}+z^{6}=1\right\} $. Note: Not how to do this problem, I not ...
0
votes
1answer
46 views

Consider the function : $L_{ij}=x_{i}\frac{\partial}{\partial x_j}-x_{j}\frac{\partial}{\partial x_i}$

Let $\Omega$ be a smooth bounded domain of $\mathbb{S}^n$, the unit $n$ sphere centered at the origin of $\mathbb{R}^{n+1}$, and consider the functions $$L_{ij}u=x_{i}\frac{\partial u}{\partial x_j}-...
2
votes
1answer
29 views

What's the Cheeger Constant of the Disc?

Context: I recently encountered the notion of the Cheeger constant in graph theory and in Riemannian geometry when planning for some intensive studies on expander graphs for the summer. To gain some ...
1
vote
0answers
45 views

Zero gradient in $L^2(M)$

I'd like to show that for $u \in L^2(M)$, for M a compact, connected Riemannian manifold, if $\nabla_g u = 0$ (i.e $\forall X$ $C^{\infty}$- vector field on $M$, $\int_M u \hspace{1mm} \text{div}_g X =...
0
votes
1answer
34 views

A corollary of Li-Yau-Hamilton estimate

Picture below is from the Hamilton's The Harnack estimate for the Ricci flow .How to get the corollary 1.2 by Theorem 1.1 ? It seemly be not immediately and hard to compute. Maybe just because I am ...
2
votes
1answer
45 views

Can we lower bound the volume of the image of a ball under a diffeomorphism?

Apologies if this question is overly simple, I'm new to differential geometry. Suppose I have two Riemannian manifolds $M_1$ and $M_2$, along with a diffeomorphism $f:M_1\to M_2$ between them. Let $d$...
0
votes
0answers
22 views

Existence of solution of integral curve on manifold

Let $M$ be an $n$-dimensional manifold and $X$ be a smooth vector field on $M$. In all books I found that the proof all uses the existence of solution of ODE in $\mathbb R^n$. I try to give an '...
1
vote
2answers
42 views

What is the Riemannian metric induced on a surface $M \subset \Bbb R^3$ by the usual flat metric?

Let $D$ be an open subset of $\mathbb{R}^3$, and $f: D \to \mathbb{R}$ be a smooth function whose gradient $ \nabla f \neq 0$ on $D$. Consider the surface $M = \{(x_1,x_2,x_3) \in D \mid f(x_1,x_2,x_3)...
1
vote
0answers
26 views

Can the rank of harmonic map decrease far from the boundary?

This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...
3
votes
0answers
27 views

Parametric surfaces in Riemannian manifolds

Let be $(M^n,g)$ a Riemannian manifold and $c: [0,l]\to M$ a geodesic with unit speed. Consider the parametric surface $f$ is $M$, given by $$f(s,t)=\exp_{c(s)}(tn(s)),$$ where $(s,t)\in [0,l]\times ...
0
votes
0answers
27 views

definition of covariant derivative (along curve)

An affine connection on a smooth manifold $M$ is a map $\nabla: \mathcal{V}(M) \times \mathcal{V}(M) \to \mathcal{V}(M)$ satisfying several properties, where $\mathcal{V}(M)$ denotes the set of smooth ...
-1
votes
1answer
22 views

Orthonormal Frame as a function

Let $M$ be a smooth manifold. We know that the frame at a point $p\in M$ can be defined as an isomorphism $f:\mathbb{R}^n\longrightarrow T_pM$. Is there a way of defining an orthonormal frame in a ...
1
vote
1answer
37 views

Strongly convex set is contractible

A subset $S\in M$ is called strongly convex, or geodesically convex, if for any $p,q\in S$ there is a unique normal minimal geodesic $\gamma$ joining $p$ to $q$, and $\gamma$ is contained in $S$. For ...
1
vote
1answer
16 views

Example of a surface with no unit-speed geodesic at all time $(-\infty,\infty)$

I am stuck on the following problem, which was given as homework. What is an example of a 2-dimensional surface in $\mathbb{R}^3$ such that it's not possible to find a unit-speed geodesic $\sigma: (-\...
2
votes
1answer
43 views

What value of $c$ makes this Riemannian metric complete?

I was given the following question in my differential geometry class. The instructor does not use a textbook, and gives only theorems and proofs with no examples, so I don't know how to do ...
0
votes
0answers
27 views

Software of symbolic computation

In Riemann geometry, there are many complex compute , for example in the picture below.If want to get 2.5.16 it needs about 3 page to compute. And it is easy to mistake because it is complex. But the ...
6
votes
1answer
104 views

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles

Show that the convex neighborhood in a Riemannian Manifold are subset contractibles (to any of their points). A subset $A$ of the differential manifold $M$ is contractible to the point $a\in A$ when,...