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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Whether there are any differential structure in the homology or homotopy of real coefficient ? [on hold]

I listen a report , the reporter says that the group of all closed curves of a Riemannian manifold has differential structure . So I guess the homology or homotopy of some suitable coefficient have ...
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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 ...
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20 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 ...
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39 views

Killing fields on product metrics

Let $(M_i,g_i)$ be Riemannian manifolds, $i=1,2$. (Save Euclidiean factors) Is it true that a Killing field $Z$ on $(M_1\times M_2,g_1\times g_2)$ will split as a sum of Killing fields $Z=X+Y$, where ...
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48 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 ...
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Gradient and Laplacian of a submanifold

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$. Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. Similarly ...
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32 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 ...
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41 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 ...
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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 ...
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17 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 ...
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10 views

solution of the Heat equation on a closed manifold

Let $\rm(M,g_0)$ be a closed Riemannian manifold and let $(g(t),\phi(t))$ is the solution of the so-called List's flow, i.e. the following system of equations $$\left\{\begin{array}{11} ...
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11 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 ...
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1answer
25 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?
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1answer
23 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 ...
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1answer
29 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 ...
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22 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 ...
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1answer
28 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 ...
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29 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 ...
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44 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: $(*) \, ...
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1answer
43 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 $t\in I$. ...
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58 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 ...
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188 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 $\mathcal T(P)$ ...
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31 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 ...
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A exercise of Riemannian geometry.

Let $(V,g)$ be Euclid vector space , $a$ is a symmetric 2-tensor , define $a^*: V\rightarrow V$ as $$ \langle a^*(X) , Y \rangle =a(X,Y) ~, ~~~~~~ X,Y\in V $$ $\langle~,~\rangle$ is inner product . ...
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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 ...
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54 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 ...
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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: ...
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14 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.
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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 ...
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16 views

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

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 ...
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1answer
43 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 ...
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1answer
32 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 ...
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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 ...
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1answer
56 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 ...
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1answer
20 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 ...
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31 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 ...
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1answer
38 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, ...
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46 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 ...
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28 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 ...
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27 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 ...
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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 ...
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1answer
27 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 ...
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42 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 ...
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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 ...
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1answer
40 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 ...
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20 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 ...
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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 ...
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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 ...
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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 ...
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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 ...