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 $-2Ric=\mathcal{L}_vg$?

$\varphi_t$ is a one-parameter group of diffeomorphisms generated by a vector field $V$ on $M$, $$ g_{ij}=\varphi_t^*g_{ij}(x,0) ~~~ \frac{\partial g_{ij}}{\partial t}=-2R_{ij} $$ How to show that $-...
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Convexity and mean curvature

Let $N$ be a Riemannian 3-Manifold with and $M \subset N$ an embedded, oriented codimension $0$ submanifold-with-boundary, bounded by a non-empty smooth subsurface $S := \partial M$. Now, with $M$ and ...
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49 views

Is the unit bundle of a Finsler vector bundle a sphere bundle?

Note: By now I have asked this question also at mathoverflow. Let $E$ be a Finsler vector bundle* of rank $k$ over a manifold $M$. Does the unit "bundle" $UE$ admits a structure of a sphere bundle? ...
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Interpreting partial derivatives $\frac{\partial f}{\partial u}$ in differential geometry

I'm reading Do Carmo's "Riemannian Geometry" and at some point he introduces the following notation: Let $A \subset \mathbb{R}^2$ be an open region bounded by a piecewise differentiable curve and $f:...
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Showing that Ricci curvature of round unit sphere $(S^n,g_0)$ is $Ric(g_0)=(n-1)g_0$

Let $g_0$ is a Riemannian metric of round unit sphere, $Ric(g_0)$ is the Ricci curvature. How to show that the $Ric(g_0)$ of round unit sphere $(S^n,g_0)$ is $Ric(g_0)=(n-1)g_0$ ? In fact, I don't ...
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How to solving the ODE $\frac{dp}{dt}=-\frac{\lambda}{p}$?

$p=p(t)$ is a function about $t$. I try to do so: $$ \frac{dp}{dt}=-\frac{\lambda}{p}~\Rightarrow ~p dp=-\lambda dt~ \Rightarrow~\int p dp=\int -\lambda dt ~\Rightarrow~ \frac{1}{2}p^2=-\lambda t ~\...
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Boundedness of curvature under Ricci flow on compact surface

Consider the Ricci flow on a compact surface of negative curvature. It has been proved by Hamilton that the flow in this case exists for all time. My question is, is the curvature uniformly bounded in ...
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Equivalent descriptions of Sobolev spaces on compact manifolds

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces. The first one, valid only for compact manifolds (because it needs to globalize ...
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Repeated application of the gradient on a Riemannian manifold

While reading about Sobolev spaces on manifolds, I encountered the following notation regarding the norm in $H^k$: $\| \nabla ^k f \|_{L^2}$. There are two questions here: 1) What kind of object id $\...
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asymptotics of distance between points on geodesics moving with constant speed

Let $(M,g)$ be a Riemannian manifold of unbounded diameter and let $\gamma_1, \gamma_2$ be two geodesics such that $\gamma_1(0)=\gamma_2(0)=x$. Suppose that $\gamma_1, \gamma_2$ are parametrized so ...
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Problems with Leibniz rule in calculating the covariant derivative of a $(1,1)-$ tensor. Where is my mistake?

Let be $$R=\sum _{\alpha, \beta} R^\alpha_\beta \frac{\partial}{\partial x^\alpha} \otimes dx^\beta. $$ I want to calculate $\nabla_\gamma(R)=\nabla_{\frac{\partial}{\partial x^\gamma}}(R).$ My book ...
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prove iff conditions on $(M_1,g_1)$, $(M_2,g_2)$ so that $\big(M_1\times M_2,g_1+g_2\big)$ satisfies $\text{Ric}^0=0$

Problem Show that a product metric $\big(M_1\times M_2,g_1+g_2\big)$ satisfies $\text{Ric}^0=0$, (where $\text{Ric}^0$ is the traceless Ricci tensor) if and only if both original metrics are ...
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100 views

Introductory book for Riemannian geometry

Could you recommend some books for Riemannian geometry to me? I am completely new to Riemannian geometry but have some very basic background for differential geometry, I want to know more about ...
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206 views

Mean curvature in terms of Christoffel symbols

The Setup is the following: Let ($N$,g) be a Riemannian $n$-manifold and $M \subset N$ an embedded codimension 1 submanifold, everything oriented. My goal is to express the mean curvature of a point $...
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60 views

A compute of Ricci Flow

Let $g(0)=g_0$ and $Ric(g_0)=\lambda g_0,\lambda\in\mathbb{R}$, the $Ric(g)$ is the Ricci curvature,$g$ is Riemannian metric. How to show that : The $g(t)=(1-2\lambda t)g_0$ is a solution of $$ \...
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Does any minimal surface which is regular has a conformal parameterization?

Since given surface is regular and minimal, I have $$H=\frac{Eg-2Ff+Ge}{2(EG-F^2)}=0$$ and $$X_u\times X_v\ne0$$ Can I derive $E=G,\ F=0$ from these conditions?
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Extend a Riemannian metric defined on the boundary.

If $M$ is a (compact) Riemannian manifold with nonempty boundary and I have a Riemannian metric defined on $\partial M$, Is it possible to obtain a Riemannian metric on the whole $M$ extending the one ...
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Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
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55 views

The Hessian at a critial point $p$

In my studies I've come across the hessian in the context of Riemannian geometry. I use the following definition of the hessian $$ H^f(X,Y)=XYf-(D_xY)f=\langle D_X(\operatorname{grad} f),T\rangle. $$ ...
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Are plane sections of codimension 1 manifolds locally minimal surfaces?

Suppose you have the following setup: Let $(N,g)$ be a (complete) Riemannian manifold and $M \subset N$ a smooth codimension 1 submanifold, $p \in M$. Then $T_pM$ is a subspace of codimension 1 in $...
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Taylor expansion of the square of the distance function

Give a smooth Riemannian manifold $(M,g)$, (i) how can one compute Taylor expansion of the square of the Riemannian distance function $d^2(x,x_0)$ at $(x',x_0)$? I've tried to use $dist=\int (g_{\mu\...
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Divergence equation on noncompact manifold.

Is it true that on any complete, non-compact Riemannian manifold $(M, g)$ there is a smooth vector field X such that $\operatorname{div} X=1$? My definition of the divergence is the coordinate one (...
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Riemmaninan Metric Outer measure?

If $M$ is a Riemmanian manifolds and $\mu$ is the Riemmanian measure thereon, then is $\mu$ equal to the metric-outer measure induced $M$'s Riemmanian metric when it is restricted to Borel sets?
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complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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Geodesics that paint curves on the projective space bundle

Let $M$ be a geodesically complete connected riemannian manifold. Let $p \in M$ be a point and $c: \mathbb{R} \to M$ an arbitrary geodesic that doesn't intersect p. Our aim is to find a "nice" map ...
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57 views

Tensor Algebra for Riemannian Geometry

I'm trying to learn a little bit about Riemannian geometry, but the books that I'm looking through seem to assume that the reader is familiar with topics such as contracting tensors, raising and ...
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52 views

Asymptotic behaviour of the length of a curve .

Let $(M, g)$ be a Riemannian manifold and $\gamma :[0, \delta] \to M$ a $C^1$-curve, $\gamma(0) = y$ and $\dot\gamma(0) \neq 0$. Then we have $$\lim_{t\to 0} \frac{d_M(y, \gamma(t))}{\int_0^t |\dot \...
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106 views

Partition of Unity for defining Riemannian metric.

Why do we need Partition of Unity for defining a Riemannian metric on a manifold ? What role does it play ?
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133 views

Local orthonormal frames

Let $M$ be a smooth manifold. For each coordinate chart $(U,\varphi)$ one can form the basis $\partial_1,...,\partial_n$ which are local vector fields defined on $U$. For fixed $x$ this forms a basis ...
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Lipschitz maps between Riemannian manifolds

Let $U \subset \mathbb R^n$ and $V \subset \mathbb R^m$ be two open subsets with $U$ convex and $f: U \to V$ a $C^1$-map. Then, using the fundamental theorem of calculus among other things, one can ...
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Two-Component Spinor Index Placement

This may ultimately be a silly question, but a pedantic mind like mine gets tied into knots over differing notation. Let $\mathbb{W}$ be a complex two-dimensional vector space which carries the ...
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Quadratic form of the square of the distance function [closed]

I'm looking for a special kind of metric (say Riemannian manifold $(M, g)$), where square of distance function $d^2:M\times M\to R$ between an arbitrary points $x$ and basepoint of $x_0$ is quadratic ...
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Sobolev space on compact Riemannian manifolds.

The Sobolev space on Euclidian space $W^{k,p}(\Omega)$ is well-known, where $\Omega$ is a subset in $\mathbb{R}^{d}$. Then, how does the Sobolev space on the compact Riemannian manifold $W^{k,p}(M)$ ...
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How does the Torsion of two vector fields act on their corresponding flows?

Let $X$ and $Y$ be vector fields defined on an open neighborhhod of a smooth manifold $M$ endowed with an (arbitrary) affine connection $\nabla$ (i'm not assuming anything apart from it being a ...
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1answer
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Klein-Beltrami Model Metric

So I just wanted to see if this was the right definition of the Klein-Beltrami model because there have been some issues with a computation. The Klein Beltrami model is the set $B(0,1)\subseteq \...
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What is the metric of the Riemann surface resulting from quotiening the upper half plane by a Fuchsian group?

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. What is the metric we get on $\mathbb{H} / \Gamma$?
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How do I show that a distribution is a function?

While reading Grigor'yan's book on the heat kernel I have encountered the following definition of a Sobolev space on a Riemannian manifold $M$: $W^2 (M) = \{ u \in W^1 (M) : \Delta u \in L^2 (M) \}...
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68 views

Clarification in the way I have to do derivatives in this problem of Riemannian geometry (Euler-Lagrange equations)

Let be $X$ a Riemannian manifold. I fix a local chart $(U,\varphi)$ If $ \phi : [a,b] \rightarrow U$ is $C^\infty$ I consider $$H(\phi )= \int_a^b k(\phi (t),\dot{\phi(t)})\; dt$$ where $$ k(\phi(t),...
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Bundle of densities

On the orientable manifold $M$ one can find a nowhere vanishing top form $\omega$: in fact the existence of such a form is equivalent to $M$ being orientable. In the nonorientable setting it is ...
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68 views

Metric compatibility, Ricci rotation coefficients & non-coordinate bases

I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric ...
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47 views

construct a connection such that a given tensor is parallel wrt it

Let $\omega$ be a symplectic form on a smooth manifold $M$. How does one construct a connection on $TM$ such that $\omega$ is parallel to it? It's easy to construct a connection on a dual bundle ...
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Linear transformation and riemannian metric tensor

When is it true that for a linear transformation $A$ and a symmetric bi-linear form $g$ (i.e. (0,2) tensor) on a riemannian manifold: $$ g( AV, W)= g(V, AW)?$$
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Deriving formula for formal adjoint

My question is in relation to the derivation of the formal adjoint for a connection $D:\Omega^{p-1}(\text{Ad}E)\rightarrow \Omega^p(\text{Ad}E)$ - I am reading through this derivation in Jost's ...
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Show that $\overline{R}(X,Y)Z=\nabla_X\nabla_YZ-\nabla_{[X,Y]}Z=-\Theta$.

Exercise I am tasked with showing the following (Exercise 2.6.13.4 in Cartan For Beginners by Ivey and Landsberg): For $X,Y,Z\in\Gamma(TM)$, define $$ \overline{R}(X,Y)Z=\nabla_X\nabla_YZ-\...
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45 views

What's the lemma's name ?And how to proof it?

I saw the lemma in a Riemannian Geometry book, but the proof is omitted ,I want to know how to proof it .Thanks.
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Are two isometric Riemannian manifolds necessarily related by Euclidean motions?

Let $(M,g), (\overline{M},\overline{g})$ be smooth Riemannian manifolds that are isometric, i.e. there is a smooth function $f: M \rightarrow \overline{M}$ such that d$f(g)=\overline{g}$. By Nash's ...
2
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1answer
58 views

Integral on Riemannian manifold

$M$ is a Riemannian manifold ,and $g_{ij}$ is Remannian metric. Let $x=(x^1...x^d)$ $(i.e. x:U_x\rightarrow R^d)$ be a local coordinates ,and $v,w\in T_pM$ with coordinate representations $(v^1...v^d)$...
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Fixing some notations about tensor calculus

I'm studing some articles about Ricci Flow and Ricci Solitons. I realize the first thing to do is choose a notation, thus I'm using the Huai Dong Cao's notation (which is the same Ricgard Hamilton's ...
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69 views

Calculation of a covariant derivative and exterior derivative

I am reading these notes on differential geometry from a course at MIT. I have been verifying the computations for myself and I have a concern about the expression for $de^k$ on the final line. When I ...
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83 views

Pseudo-Riemannian Metric on Manifold

In Riemannian geometry, we have Proposition Any Manifold has a Riemannian metric. However, we cannot place the proof on pseudo-Riemannian situation because we do not hold the signature on ...