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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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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1answer
561 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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1answer
94 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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1answer
87 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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186 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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142 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 ...
2
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1answer
326 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 ...
2
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1answer
217 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
295 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 ...
3
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1answer
46 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 ...
2
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1answer
194 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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1answer
150 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
109 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, ...
2
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1answer
881 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 ...
6
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2answers
288 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
62 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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219 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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83 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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1answer
88 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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0answers
109 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 ...
6
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2answers
298 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 ...
4
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1answer
175 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 ...
3
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1answer
307 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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1answer
239 views

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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2answers
129 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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1answer
105 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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1answer
280 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
85 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
164 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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0answers
53 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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0answers
62 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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4answers
2k views

Proving the symmetry of the Ricci tensor?

Consider the Ricci tensor : $R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho} -\partial_{\nu}\Gamma_{\rho\mu}^{\rho} +\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda} ...
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1answer
228 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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0answers
307 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)} ...
3
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1answer
139 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 ...
4
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1answer
219 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 ...
4
votes
1answer
386 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 ...
4
votes
1answer
288 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 ...
6
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2answers
344 views

Does the Levi-Civita connection determine the metric?

Can I reconstruct a Riemannian metric out of its Levi-Civita connection? In other words: Given two Riemannian metrics $g$ and $h$ on a manifold $M$ with the same Levi-Civita connection, can I conclude ...
4
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0answers
238 views

geometric meaning of Ricci-flatness

What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
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0answers
104 views

A question from Hamilton's Ricci Flow book by bennett chow

On page 3 of the book before exercise 1.2, has written: "torsion free is a compatibility condition with the differentiable structure". I correctly do not understand how torsion-free condition results ...
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1answer
106 views

Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
3
votes
2answers
134 views

Minimal length of non-contractible loops

Not self-intersecting loops on a connected closed orientable smooth surface $S$ must have a minimal length not to disconnect it, e.g. the equators of a torus. "Not to disconnect" is - on such surfaces ...
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1answer
98 views

Surfaces without conjugate points

I'm trying to understand some aspects of the geodesics of surfaces without conjugate points and the following question came out: consider the hyperbolic space and two geodesics wich are asymptotic to ...
4
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2answers
267 views

Property of normal coordinates

Let $M$ be a Riemannian manifold and $\nabla$ the Levi-Civita conection. I need to prove the following. Let $B$ be an open ball of radius $r$ in $T_pM$ such that $\left.exp_p\right|_B$ be a ...
4
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1answer
225 views

The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)

In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it. Let $\gamma: ...
5
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1answer
243 views

Relating volume elements and metrics. Does a volume element + uniform structure induce a metric?

AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way ...
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1answer
166 views

Show that the vector field $\operatorname{grad}f$ is smooth

Let $M$ be a Riemannian manifold and $ f:M\rightarrow\mathbb{R}$ be a smooth function. Define a vector field $\operatorname{grad}f$ in $M$ as $$\langle\operatorname{grad}f,\,V\rangle=df(V)$$ for all ...
7
votes
1answer
396 views

Vector field and integral curve

Let M a riemaniann manifold, $V$ a vector field in M, and $\phi_t$ the respective flow Let $p\in M$, and $\gamma$ the integral curve of $V$. Show that for all $v\in T_pM$ ...
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1answer
45 views

Question about self homeomorphism of $\mathbb C\mathbb P^2$

Can anyone give me any idea about how to show that: any self homeomorphism of $\mathbb C\mathbb P^2$ is orientation preserving? Thanks.