(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

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Using Gauss-Bonnet to prove that geodesics have at most one point of intersection

Given an oriented Riemannian manifold $(M,g)$ of dimension $2$, such that $M$ has negative Gaussian curvature everywhere and $M$ is diffeomorphic to $\mathbb R^2$, I'm looking for a way to show that ...
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89 views

Prove Green formula

Let $(M^n,g)$ be an oriented Riemannian manifold with boundary $\partial M$. The orientation on $Μ$ defines an orientation on $\partial M$. Locally, on the boundary, choose a positively oriented ...
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72 views

How to conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$?

Can anyone explain to me how I can conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$ by using integration by parts and $\langle f_1 ,f_2 \rangle_\mu:=\int_M f_1 f_2 d\mu$? Where $M$ is a ...
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151 views

Exponential Map

this seems to be an easy question but I'm stuck anyway. Let $\Gamma$ be a submanifold of a Riemannian manifold $(M,g)$. Let further $U$ be a coodinate neighborhood of $\Gamma$ such that a point ...
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58 views

structure of Riemannian manifold of isometries from C^n to C^m

Does anyone know a reference which gives the properties (geodesics, geodesic distance, etc) of the Riemannian manifold of isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>n$, which map zero ...
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91 views

Solution to $\Delta_g u = \delta-1$ on a 2-sphere.

Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
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145 views

Is a Riemannian metric positive definite or positive semidefinite?

From Wikipedia The Fisher information matrix is a N x N positive semidefinite symmetric matrix, defining a Riemannian metric on the N-dimensional parameter space, But a Riemannian metric is ...
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142 views

Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
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110 views

Zeta Regularized Determinant of Laplacian

Can anyone point me to a resource where the zeta regularized determinant of the Laplacian is explicitly computed for simple two dimensional surfaces, say a rectangle or torus or cylinder?
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101 views

Geodesics and Christoffel symbols

If these are satisfied then we are on a geodesic. Do I just need to plug in the condition given about the christoffel symbols and then see that the equations are allways fulfiled as long as $v=at+b$ ...
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384 views

Ricci tensor for a 3-sphere without Math packets

Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
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76 views

Orientability of $P_{\bf R}T{\bf RP}^{2n}$

I know the following fact : (1) $ {\bf RP}^{2n}$ is non-orientable. (2) $ {\bf RP}^{2n-1}$ is orientable. (3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable. (4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
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494 views

Gradient in Riemannian manifold

I have a calculation involving a gradient and a parametrization, but I haven't been able to find out the relation between them. Let me explain. Let $f:X↦R$ be a smooth function and $\mathrm{grad}f\in ...
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1answer
44 views

what is the inner product appeared in front of the integral?

Given a compact manifold with a Riemannian metric $g$, we define the total scalar curvature by $$E(g)=\int_M RdV$$ Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
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377 views

Riemannian measure and Hausdorff measure in a general Riemannian Manifold

Let $ M $ be a Riemannian manifold and let $ \mu $ be its Riemannian measure. This is the measure obtained by Riesz reprersentation theorem such that for every continuous function with compact support ...
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70 views

Meaning of modulo diffeomorphism

I faced this sentence: we consider the space of Riemannian metrics modulo diffeomorphism and scaling. Can anyone explain to me what is the meaning of modulo diffeomorphism and scaling? Thanks!
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76 views

Killing Vector Field determined by one point

I am trying to prove that if $X$ is a Killing vector field on a connected Riemannian manifold $(M,g)$ (i.e. $\mathfrak L_X g = 0$), then $X$ is determined by $X_p$ and $\nabla X|_p$ for any point $p ...
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172 views

Hopf's theorem on CMC surfaces

I got stuck reading the proof of the following theorem: Theorem (Heinz Hopf) Let $X: S^2\to \mathbb R^3$ be a constant mean curvature immersion. Then $X(S^2)$ is a round sphere. Proof: Let ...
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126 views

Energy functional

During my study on Ricci Flow I faced some functional known as enery functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...
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274 views

Contraction of the second Bianchi identity

The second Bianchi identity is $${R^a}_{b[cd;e]}=0$$ And contracting it with respect to $a$ and $e$ we get $${R^a}_{b[cd;a]}=0 \Leftrightarrow $$ $${R^a}_{bcd;a}+R_{bc;d}-R_{bd;c}=0$$ What I don't ...
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155 views

Gradient of a functional

Given a compact manifold with a Riemannian metric $g$, we define the total scalar curvature by $$E(g)=\int_M RdV$$ Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
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2answers
259 views

Justification for this manipulation in a proof of the first variation of energy formula

As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in these ...
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1answer
40 views

Cut locus of $\mathbb{CP}^n$

I can show that the cut locust of some $p\in\mathbb{RP}^n$ is just a copy of $\mathbb{RP}^{n-1}$ coming from an equatorial $S^{n-1}$ sphere under the projection $S^n\mapsto\mathbb{RP}^n$. I know that ...
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163 views

Volume element of the Sphere

If we consider the sphere on $E^3$ with Riemannian metric $G=dx + dy + dz$ then transforming to spherical coordinates we get $G=R^2 d{\theta} +R^2 sin(\theta) d{\phi}^2 $. Hence the volume form is ...
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127 views

Volume of a 3D sphere of radius $R$ using Riemannian metric in stereographic coordinates

The question is pretty much in the title. We were also given the hint that it could be useful to use spherical coordinates when calculating the integral (the actual answer is not required, just its ...
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1answer
78 views

Non-linear transformation preserving stereographic Riemannian metric on the sphere of radius R

I have been given a the Riemannian metric of a sphere of radius R in stereographic coordinates: $$G=4R^4\frac{du^2+dv^2}{(R^2+u^2+v^2)^2}.$$ I have shown that this metric is preserved under rotation, ...
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539 views

Quotient theorem for tensors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
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183 views

Metric on Steifel and Grassmannian manifolds generalizing Fubini-Study

If $F$ is $\mathbb{R}, \mathbb{C}$, or $ \mathbb{H}$, the Grassmannian manifold $G_k(\textbf F^n)$ is the space of all $k$ dimensional subspaces of the $n$ dimensional vector space $F^n$. The Stiefel ...
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1answer
54 views

Confusion regarding uniqueness of Levi-Civita connection

Assuming a Levi-Civita connection exists it is uniquely determined. Using $\nabla g = 0$ and the symmetry of the metric tensor $g$ we find: $ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(Y,X)) = ...
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196 views

Riemannian curvature and its application on covariant derivative of tensors

This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
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76 views

Existence of Solution: Embedding from 2D Euclidean space to a circle

Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
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55 views

Question regarding Nash-Kuiper embedding theorem

In Wikipedia description of Nash-Kuiper theorem, it says: Let $(M,g)$ be a Riemannian manifold and $f: M^{m} \rightarrow \mathbb{R}^n$ a short $C^{\infty}$-embedding into Euclidean space ...
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99 views

Reason for defining Riemannian curvature tensor and torsion tensor in particular way

I saw how Riemannian curvature tensor and torsion tensor are defined, but I am not sure why they are defined that way. In 3-dimensional euclidean space with ordinary multivariable calculus, the ...
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203 views

Problem about Ricci Flow

On page 12 of "Lectures On Ricci Flow" by Peter Topping is written: In two dimensions, we know that the Ricci curvature can be written in terms of the Gauss curvature $K$ as $Ric(g) = Kg$. Working ...
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144 views

covariant derivative vs. exterior derivative

I have the following question. Let $M$ be a Riemannian manifold with metric $g$ and $\nabla$ the Levi-Civita connection. Let furthermore $\alpha \in \Omega^{k}(M)$ be a $k$-form such that $\nabla ...
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189 views

the Ricci curvature in two dimension

In two dimensions, we know that the Ricci curvature can be written in terms of the Gauss curvature K as $Ric(g) = Kg$. Can anyone prove this? Sorry if the question is too trivial :).
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About Sectional Curvature

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
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109 views

Showing that the riemanian metric $\frac{g}{\sqrt{x^2+y^2+z^2}}$ is complete

I would like to show that a certain Riemannian metric defined on $\mathbb{R^3}$ is complete. The metric is given in the following sentence from this article (pg 160): ... a Riemannian metric on ...
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80 views

Elementary definition: what's a parallel volume-form?

This is a very elementary question, What is the definition for a volume form (or $n$-form) to be parallel with respect to the metric? To find out more about the concept, what kind of topic do I need ...
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221 views

Curvature of a metric defined on an open disc in $\mathbb{R}^2$

Let $D$ be an open disc centred at the origin in $\mathbb{R}^2$. Give $D$ a Riemannian metric of the the form $(dx^2+dy^2)/f(r)^2$, where $r=\sqrt{x^2+y^2}$ and $f(r)>0$. Show that the curvature of ...
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1answer
83 views

Difference between “Live” and “Define”

In many mathematical text to determine an object on manifold, the verbs "live" and "define" are used. I'm interested to know whether there is a difference between the concepts of "to define" and "to ...
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1answer
141 views

Parallel transport for a conformally equivalent metric

Suppose $M$ is a smooth manifold equipped with a Riemannian metric $g$. Given a curve $c$, let $P_c$ denote parallel transport along $c$. Now suppose you consider a new metric $g'=fg$ where $f$ is a ...
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49 views

A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped

On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which ...
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53 views

Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov

Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
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206 views

Formula for Gaussian curvature in terms of unit tangent vector fields?

Let $X\in\mathbb{R}^3$ be a surface with a local geodesic polar parametrisation with first fundamental form $du^2+G(u,v)dv^2$. How do we define unit tangent vector fields $e_1$, $e_2$ on $X$, forming ...
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228 views

Difference between parallel transport and derivative of the exponential map

Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then we have to ways to map $T_pM$ to $T_{c(t_0)} ...
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126 views

Zeros of the second fundamental form

Let $ f:M \rightarrow N $ be a minimal immersion (of arbitrary codimension or an hypersurface if it is necessary) and let $ |A| $ be the norm of its second fundametal form.If $ A $ is not identically ...
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164 views

Elliptic equation on riemannian manifolds

Let $ M $ be a compact Riemannian manifold with or without boundary) and let $ \Delta $ be the metric laplacian. I want to study the differential operator $ -\Delta +q $ where $ q $ is a smooth ...
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279 views

Why do we need Lie derivative?

If a manifold is equipped with Levi-Civita connection, Why do we need Lie derivative? In Euclidean space to calculate directional derivative of a vector field V along W, we parallel transport V along ...
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233 views

A question about concept of pushforward

In An Introduction to Smooth manifolds by Lee is written: for any smooth vector fields V and W on a manifold $M$, let $\theta$ be the flow of $V$, and define a vector $(\mathcal{L}_v W)_p$ at each ...