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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Basic question about the metric tensor

So I am reading Barrett O'Neil's Elementary Differential Geometry, which defines the metric tensor $g_p$ on a surface $M$ to be a bilinear symmetric positive-definite function on the set of ordered ...
3
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51 views

ALL Orthogonality preserving linear maps from $\mathbb R^n$ to $\mathbb R^n$?

That is we have a linear transformation, i.e. an $ n\times n $ matrix $A$, such that for every pair of vectors $ v $ and $ w $ we have $$ \langle v,w\rangle=0 \ \ \ \implies \ \ \ \ \ \langle Av,...
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1answer
29 views

Computing the “Mean Value” of a Point Sample From an Arbitrary Manifold

A friend of mine noticed that taking the "mean" of two points on the circle isn't as easy as just computing the arithmetic mean of their arguments: If one point has argument $-3.13$ radians and one ...
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28 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
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1answer
25 views

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible. The problem is from the following material. It contends that the result is by standard Morse theory....
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47 views

Kato's inequality

Let u be a smooth function defined in a Riemannian manifold $(M,g)$. The well known Kato's inequality states $$|∇|∇u||^2≤|∇^2u|^2$$ where $∇^2$ represents the Hessian operator of $M$. I would like ask ...
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24 views

a algebraic concept in reimannian geometry?

let $(M,g)$ is a $C^{\infty}$ manifold with product form of $M=R \times \Theta $ the for each element $\varepsilon (r,\theta )\in M$, the tangent space $T_{\varepsilon}M$ is natural isomorphic to $...
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1answer
50 views

Whether there is easy way to compute $R_{ij}=\frac{1}{2}Rg_{ij}$ in 2-dimension

In 2-dimensional Riemann manifold ,Ricci curvature is given by $$ R_{ij}=\frac{1}{2}Rg_{ij} $$ My PDE teachers teach me to compute it by the way. $$ R_{11}=g^{ij}R_{1i1j}=g^{22}R_{1212} \\ R_{12}=g^...
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31 views

Riemann Curvature tensor for surfaces

Let $M$ be a regular surface on $\mathbb{R}^3$. I am trying to express the Riemann's curvature tensor: $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ respect $R(\vec x_i,\vec x_j)\vec ...
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What do I need to review for Riemannian Geometry?

Well, I have been about two years without studying almost anything. I am going to start a thesis about three dimensional spaces (need to understand and explain their isometries, curvature, geodesics), ...
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1answer
21 views

Why the well-defined of Gauss map depends on surface is orientable?

Let $S$ is a surface. Define a mapping $g:S\rightarrow S^2\subset R^3$ of $S$ into the unit sphere $S^2$ , associating to every $p\in S$ a unit vector $N(p)\in S^2$ normal to $T_pS$. Why the well-...
2
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2answers
57 views

Canonical notion of parallel transport

I have a "What is the right search term?" style question: Suppose $S\subset\mathbb{R}^3$ is a surface and that we are given two points $x,y\in S$. Furthermore, take $v_x\in T_x S$ to be a tangent ...
1
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0answers
37 views

How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$?

$\nabla$ is Riemann connection and $R_{ij}=g^{kl}R_{ikjl}$. How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$ ? Or generate commutator of generate ...
1
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1answer
20 views

Question about homogeneity of the geodesic

I have the following questions about the Homogenity of geodesic: Let $\gamma:(-\delta,\delta)\to M$, where $t\to\gamma(t,q,v)$ is a geodesic, then $\gamma:\left(-\dfrac{\delta}{a},\dfrac{\delta}{a}\...
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31 views

Dirichlet problem for a ball in a Riemannian Manifold

I'm trying to work through the 2nd chapter "Volume comparison" of Jeff Cheeger's "Degeneration of Riemannian Metrics under Ricci Curvature Bounds". But with the last section I have some problems. ...
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1answer
27 views

Is the set $\bigsqcup_{p\in M} \{v\in T_pM: |v|_g< r_p\}$open in $TM$?(where $r_p$ the injectivity radius at $p$)

Let $(M,g)$ is a Riemannian manifold. (1)If $D_p$ is the largest domain on which $\exp_p$ can be a diffeomorphism, then is the set $$D=\bigsqcup_{p\in M} D_p$$ open in $TM$? (2)Likewise, if we denote ...
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1answer
60 views

Divergence of vector field on manifold

This is a follow-up question to the one I made here. On the wiki page, the divergence of a vector field $X$, denoted $\nabla\cdot X$, is defined as the function satisfying $\left(\nabla\cdot X\right)\...
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2answers
96 views

Show that , for all $(s_{0},t_{0})\in [0,1]\times [0,a]$, the curves $s\to f(s,t_{0})$, $t\to f(s_{0},t)$ are orthogonals.

Let $f:[0,1]\times [0,a]\to M$ a parameterized surface such that for all $t_{0}\in[0,a]$, the curve $s\to f(s,t_{0})$, $s\in [0,1]$, is a parameterized geodesic by arc lenght , orthogonal to the curve ...
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1answer
24 views

Lie derivative of two differnt size related tensors

Let $\bar{M}=I\times M$ be a pseudo-Riemannian manifold equipped with metric $\bar g=-dt^2\oplus f^2g$ where $(M,g)$ is a Riemannian manifold, $I$ is an open connected interval and $f$ is a positive ...
2
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1answer
48 views

Affine manifolds which are *not* locally symmetric

Let $M$ be a manifold equipped with an affine connection $\nabla$. By standard theorems of differential geometry we have convex normal coordinate balls around every point in which geodesics are axis. ...
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27 views

Metric evolving under Ricci flow with nonnegative scalar curvature is shrinking?

Let $g_{ij}(x,t)$ be a complete solution of the Ricci flow on a surface with bounded and nonnegative scalar curvature ,why the metric is shrinking ?
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11 views

All movements that preserve the interval

Given the Minkowski space with the usual metric, I have to find all the movements that preserve the interval. I have been able to prove that the Lorentz Transformations are invariant but, ...
1
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1answer
47 views

Understanding wedge products for differential forms

I am trying to understand the derivation of coordinate expression for the Laplace-Beltrami operator (wiki here). The Wikipedia page says that $\nabla\cdot X$ is an operator mapping a function to a ...
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0answers
27 views

What is $P\times_G E$?

I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...
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1answer
42 views

Two proof of Petersen's 'Manifold'

Picture below is from the 5 page of Petersen's Manifold. First, why diffeomorphism is forced to be a lieanr isomorphism? The define of diffeomorphism accords to Wiki. Second , what space the point $...
3
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1answer
44 views

The continuity of injectivity radius

Let $M$ be a Riemannian manifold. $r:M\to [0,+\infty]$ denotes the function assigns to $p\in M$ the injectivity radius $r_p$ of the exponential map $\exp_p$. Is this function $r$ is continuous or ...
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0answers
17 views

Proof of an equation of Contact Riemannian metric structure.

Let $(M,g, \eta,\xi,\phi)$ be contact metric structure and $\{e_0=\xi,e_i,\phi e_i\}$ be a local orthonormal frame so-called $\phi$-basis. How to prove the following equation: $$g\big((\nabla_Xh\phi)...
4
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2answers
67 views

Proving the Ricci identity

I'm trying to prove the Ricci identity Let $Z^a$ be a vector field, $R^a_{\,bcd}$ the Riemann curvature tensor and $\nabla$ a torsion-free connection. Then: $\nabla_c\nabla_dZ^a-\nabla_d\nabla_cZ^...
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25 views

Why does the Yamabe problem only consider compact manifolds

Why is the assumption of compactness so important to the statement of the Yamabe problem? What goes wrong if the manifold is not compact?
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1answer
45 views

Why tensor product is used in local expression of Riemannian metric? [closed]

g = $g_{ij} du^i \bigotimes du^j $ Why tensor product is used here. Please explain.
0
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1answer
44 views

Convex subsets of pinched Hadamard manifolds

Let $X$ be a pinched Hadamard manifold (in my particular case, $X=\mathbb H^n$ is the $n$-dim. hyperbolic space) and $N$ be a closed (edit : open) convex subset of $X$. Is it true that $N$ is also a ...
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36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
2
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1answer
25 views

Let $(M,g)$ be a riemannian manifold. Show that the topology induced by the metric coincides with the topology of $M$. [duplicate]

I know that I have to show that $M$ with these two notions of topology have the same open. To do this, I have to take elements of the basis of any topology. But I am failing in choose that opens for ...
0
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1answer
44 views

Why is a metric?

I have a question about tensors and metrics: Let $M=\{(t,x,y,z)\in \mathbb{R}^4: t>-1 \}$ and let $g=(1+t)dtdx+dy^2+dz^2$ Show that g is a metric on $M$. I did the next, I have the basis $\{ \...
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26 views

Computing of proof of Li-Yau estimate

I try to compute the red line in picture below: \begin{align} \Delta(\partial_tL) +R\Delta L +\partial_t R &=\Delta(Q+|\nabla L|^2)+R(\frac{\Delta R}{R}-\frac{|\nabla R|^2}{R^2}) + \Delta R +R^2 ...
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17 views

Nonstandard support functions for the Busemann function

Let $(M,g)$ be a $n$-dimensional complete Riemannian manifold. Assume that $M$ contains a ray $\gamma : [0, \infty) \to \mathbb{R}$. Let $b_\gamma$ be the associated Busemann function, i.e. $$ b_\...
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2answers
60 views

Equivalent definition of metric compatibility for a connection: what does $\nabla g$ mean?

So the definition I know for metric compatibility is: $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ which make sense, as $g(Y,Z)$ is a smooth function from the manifold to reals and we think of $X$ as ...
1
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1answer
16 views

Let $\alpha, \beta:I\to M$, two geodesic. If exist a number $a\in I$ such that $\alpha'(a)=\beta'(a)$, then $\alpha=\beta$

Let $\alpha, \beta:I\to M$, two geodesic define on diff. manifold $M$ and $I$ connected. If exist a number $a\in I$ such that $\alpha'(a)=\beta'(a)$, then $\alpha=\beta$. My approach: Define $\gamma:...
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1answer
20 views

Show that exist a unique field $G$ on $TM$ whose paths are of the form $t\to (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$.

Show that exist a unique field $G$ on $TM$ whose paths are of the form $t\to (\gamma(t),\gamma'(t))$, where $\gamma$ is a geodesic on $M$. My approach: Suppose such field actually exist, consider a ...
2
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1answer
27 views

Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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12 views

Invariant of support function and support point under parallel translation

Picture below is from the 222 and 220 page of this paper,why the support function and support point is invariant under parallel ?
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1answer
25 views

On a proof that the metric volume form is parallel wrt to the Levi-Civita connection

In the context of (semi-)Riemannian geometry, the following fact is well-known: if a (semi-)Riemannian manifold $(M,g)$ is oriented, then the unique volume form $\epsilon = \mathrm{vol}_g$, induced by ...
2
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1answer
31 views

Lemma characterizing second fundamental form, do not understand step

Consider an excerpt of a lemma and part of its proof from a Riemannian geometry textbook. Lemma. The second fundamental form is independent of the extensions of $X$ and $Y$; bilinear ...
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1answer
13 views

Euler-Lagrange implies that geodesics has constant norm of velocity

I am trying to prove that geodesics has norm of velocity constant. To do this, I applied that Euler-Lagrange equation to $$S(\gamma,\gamma') := \int_{0}^1 \|\gamma'(t)\|^2dt.$$ The Euler equation ...
6
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1answer
106 views

How much classical geometry must a geometer know?

From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know): Absolute Euclidean Non-Euclidean ...
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+50

An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that $\gamma'_1(1)...
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1answer
10 views

Are segment domains closed?

Let $M$ be a complete Riemannian manifold. Its segment domain is defined by: $$ \mathbf{seg}(p)= \{v\in T_pM: \exp_p(tv):[0,1] \to M \textit{ is a segment} \ \ \} $$ (Note: "segment" has many ...
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26 views

Full definition of Rough Laplacian and induced formal adjoint of covariant derivation?

Can everybody give a good reference for full definition of Rough Laplacian of tensor field and induced formal adjoint of covariant derivation on a riemannian manifold? I find some equivalent ...
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39 views

Why Bi invariant metric on noncompact lie group doesn't exist??

In the book "Lectures on Differential Geometry" by Sternberg page 233 "Given a representation,p, of a Lie group G (in particular the adjoint representation) on a vector space F, if p(x) is compact ...
2
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2answers
35 views

Why image of curvature is a Lie subalgebra?

In the red line of picture below, why it is Lie algebra ? $M_{\alpha\beta}$ is the Lie bracket ? But $M_{\alpha\beta}$ is symmetric . Picture below is from the 216 page of this paper.