(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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Of the three tensors, Riemann Tensor, Ricci Tensor, and Ricci Scalar, which ones are only zero in a flat metric?

I think that the Riemann tensor is zero only in the presence of a flat metric. However, the Ricci Tensor and the Ricci Scalar, are unknown to me, whether they are zero only in the presence of a flat ...
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13 views

Under what conditions does a one variable off diagonal component make a curved metric irreducible into a diagonal curved metric?

I am currently working with the following metric \begin{align} ds^2 = P(r)dt^2-2C(r)dtdr-Q(r)dr^2-r^2d\theta^2-\sin^2(\theta)r^2d\phi^2 \end{align} and I am attempting to solve a problem concerning ...
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162 views

What is the scalar product of tensors?

Given there a vector space $V$ with a scalar product $g(v_1,v_2)$ on it, what is the scalar product on, say, $V \otimes V^*$ ? According to Jeffrey Lee's "Manifolds and Differential Geometry" (see ...
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10 views

Question about Schur's theorem..

I understand that if a manifold is isotropic at any point, then the manifold has constant scalar curvature and the following identity holds: $$R_{abcd}=\frac{R}{n(n-1)}(g_{ac}g_{bd}-g_{ad}g_{bc})$$ ...
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Boothby's exercise $I.3.1$ on showing that a subset of $\mathbb{R}^{2}$ is not a manifold

First time here, so sorry for the rookie mistakes. I'm a $4^{th}$ year physics student taking Riemannian Geometry so my background on the subject is very small. I'm trying to solve Boothby's exercise ...
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36 views

Computing the Fubini-Study metric

I would like to compute the Fubini-Study metric $g_{FS}$ for $\mathbb{CP}^{n}$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration. I tried to compute on ...
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12 views

“Non-Euclidean Geometry” (BONOLA, R.) appendix V paragraph 5

I'm trying to understand the appendix V paragraph 5 of "Non-Euclidean Geometry" (BONOLA, R.) Where an euclidean geometry is explained where every line is a circle trough a common point. My ...
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42 views

If $\operatorname{div} X = 0$ what can be said about $X^\flat$?

If vector field $X$ is divergent free $$\operatorname{div} X = 0$$ what are the properties of a corresponding covector field $X^\flat$ (via musical isomorphism with a metric $g$)? Are there some ...
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2answers
51 views

Christoffel's symbols for a dual connection

Suppose that $\Gamma^{\beta}_{i\alpha}$ are Christoffel symbols for a connection with respct to a (local) basis $\{E_1,...,E_n\}$. I tried to prove that the Christoffel symbols for a dual connection ...
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441 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
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About Gauss-Bonnet Theorem

The Gauss–Bonnet theorem say that: If $\Sigma \subset M =\mathbb{R}^3$ is a compact 2-dimensional Riemannian manifold without boundary, then $$ \int_{\Sigma} K = 2\pi\chi_{\Sigma}$$ where $K$ is the ...
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95 views

Confusion regarding Riemann normal coordinates

I'm trying to understand Riemann normal coordinates. This "simple" example using the surface of a unit sphere is from http://www.maths.bris.ac.uk/~macpd/gen_rel/snotes.pdf (p26). The “north pole” ...
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44 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
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1answer
36 views

When to take derivative with respect to distance?

I had a previous question about the divergence in spherical coordinates and using the usual formula found on wikipedia "List of formulas in Riemannian geometry" I could not get the correct form of the ...
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26 views

Relationship between differentiation and integration of vector fields?

Let $V\in\Gamma(T\mathbb{R}^n)$ be a vector field and $\gamma:[a,b]\to \mathbb{R}^n$ a curve. Let $\nabla$ be the Euclidean connection, i.e. $\nabla_XY=XY^k\frac{\partial}{\partial x^k}$. We have a ...
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22 views

How to compute the derivative of this functional on a manifold?

I'm a little puzzled by the following computation. Suppose $\Sigma$ is a compact two-dimensional Riemannian manifold and $M$ is any Riemannian manifold. Let $\Omega$ be some space of functions from ...
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1answer
52 views

Why are parallel vector fields called parallel?

In Lee's "Riemannian Manifolds: An Introduction to Curvature" given a curve $\gamma:[a,b]\to M$ and a tangent vector $V_0\in T_{\gamma(t_0)}M$, where $t_0\in [a,b]$, there is a drawing of the parallel ...
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30 views

what is wrong with my (too strong to be true) generalization of a Gromov result?

In his paper "Volume and bounded cohomology", page 59 (267), Gromov proves the following result: "Let $V$ be a smooth $n$-dimensional manifold, and let $P$ be a piecewise smooth polyhedron of ...
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1answer
33 views

Riemannian geometry algebra

Is this derivation correct? $$ R^{ab}_{;a}=0 $$ $$ g_{ac}g_{bd}R^{ab}_{;a}=0 $$ $$ (g_{ac}g_{bd}R^{ab})_{;a}=0 $$ $$ R_{cd;a}=0 $$ And does that mean I now have $n^3$ equation as opposed to $n$?
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17 views

computing the components of $f^*g_N$

Let $M$ and $N$ are to compact complex manifolds of dimensions $m$ and $k$ respectively, and $f:M\to N$ is a holomorphic map then how can we compute the components of $f^*g_N$
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mean curvature of a nonlinear parabolic PDE

I want to compute the mean curvature of a reaction diffusion equations of the form $$ u_t=u_{xx}-u^2 $$ Any suggestions is appreciated!
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58 views

Space of oriented lines in $\mathbb{R}^{n+1}$ as symplectic quotient.

I've been working out a nice example of symplectic reduction, and have come to a solution only after quite a lot of effort. So I was wondering if anyone knew a more straightforward route to the ...
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1answer
30 views

Connected and simply connected neighborhoods

Suppose that $E \to M$ is a (smooth) vector bundle over smooth manifold $M$. One can find the covering $\{U_i\}_i$ with the property that $E|_{U_i}$ is trivial vector bundle. The prooblem is the ...
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2answers
248 views

Hilbert theorem and constant negative curvature surfaces

Let us consider the tractroid (pseudosphere) obtained by rotation from the tractrix curve. The surface is not defined on the "big rim", so it is not a complete set. Hilbert's theorem states that there ...
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54 views

Tangent space of loop space.

Let $\Omega$ be the path space of a riemannian manifold $M$. I have to define the tangent space of $\Omega$ in a path $\omega$, that I denote with $T_p \Omega$. I think that this space is the vector ...
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64 views

Why Riemannian metrics have to be smooth?

Why do Riemannian metrics have to be smooth? Can you give an example of a smooth curve with a none smooth metric and show me what possibly will go wrong if our metric is not smooth?
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32 views

Holonomy of H^{n}

I am trying to show that $Hol_{p}(H^{n})=SO(n)$. I know that Iso$_{p}=SO(n)$. From here can I conclude that $Hol_{p}(H^{n})=SO(n)$? For $S^{2}$ if we have two vectors $u,v$ at north-pole $N$ then let ...
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Ovals of constant $ k_g$ on constant $K$ surfaces

Prove that: Constant geodesic curvature lines on constant Gauss curvature surfaces are closed Ovals/Loops. Find perimeter/length of this Oval/Loop in terms of $ k_g$ and $K$ I believe the proof ...
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25 views

A Couple Formulas in Besse's “Einstein Manifolds”

In Besse's "Einstein Manifolds," Chapter 6D, there are 2 formulas which I am interested in, which apply to compact Riemannian $4$-manifolds: ...
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1answer
41 views

Why are there infinitely many connections on a Riemannian manifold?

I've just started learning some Riemannian manifold stuff, and I'm getting confused about the concept of connection. A connection $\nabla: \Gamma(T\mathcal{M})\times \Gamma(T\mathcal{M}) \rightarrow ...
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Does local reducibility imply global reducibility of universal covering?

Let $M$ be a locally reducible Riemannian manifold, that is, for any $p \in M$, we can find an open set $U$ around $p$ and two Riemannian manifolds $X$ and $Y$ such that $U$ is isometric to $X \times ...
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Simple question on symmetric tensors 2

This question is related to this one Simple question on symmetric tensors. To prove that a vector field $Z$ is Killing, we use the identity $$0=(L_Zg)(X,Y)=g(X,\nabla_YZ)+g(\nabla_XZ,Y)\ \ \ \forall ...
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1answer
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Ellipticity of Ricci tensor, does it depend on coordinates?

Well, I am afraid this is a silly question because I know the answer must be 'yes, it does'. But I don't see why. I put the problem in context. The ricci tensor can be regarded as a differential ...
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1answer
19 views

Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
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32 views

clear statement about the relation between curvature and rotating vectors along a loop

in my research, I need to understand the relation between the formal definition of Riemann curvature tensor ($R_{jkl}^i$)and the ``vector rotation" approach: parallel transport a vector along a loop, ...
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37 views

Affine connection

The affine connection is not in general defined uniquely by the smooth structure and the Riemannian metric. Can you give some demonstration with some examples?
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127 views

How can I compute the area of a geodesic triangle?

How can I compute the area of a geodesic triangle in a Riemannian 2-manifold? If the Gauss curvature $K$ is constant and positive I can take the Gauss-Bonnet theorem to obtain ...
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1answer
48 views

Hessian is proportional to the metric everywhere

Let $(\Omega^{n+1},g)$ be a compact Riemannian manifold with smooth boundary. Let $f\in C^{\infty}(\bar{\Omega})$ satisfies $\operatorname{Hess}f=\frac{1}{n+1}g.$ Suppose the minimum of $f$ occures at ...
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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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Yarn-like functions

When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than ...
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Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
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1answer
95 views

The relation between geodesics and distances on a Riemannian manifold

My question is about computing the distance between two points in a Riemannian manifold. Suppose that $(M,g)$ is compact so that it is geodesically complete and geodesically convex. Let ...
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25 views

Metric Tensor Antisymmetry

The metric tensor on a Riemannian manifold is given as a symmetric $n \times n$ symmetric matrix (so $g_{ij} = g_{ji}$). Is there an intrinsic reason for this symmetry? Why can't it be antisymmetric ...
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Definition of a lipschitz 1-form on a manifold

What is the definition of a Lipschitz-regular 1-form on a riemannian manifold?
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29 views

In uniform circular motion in R^2, is acceleration in the normal bundle?

In physics we learn that accleration is a vector quantity parallel to the radius and orthogonal to the velocity. With the embedding $\mathbb{S}^1 \hookrightarrow \mathbb{R}^2$ and the induced ...
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30 views

Comparing PDE solutions for different Riemannian metrics

I'm looking for the approach to compare PDE solutions on the Remannian manifolds when those solutions are obtained under two different metrics. To be more specific, suppose we have two Riemannian ...
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51 views

geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
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The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
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137 views

How to define intrinsic curvature?

I have been exploring differential geometry slightly... And i'm trying to grasp how to define intrinsic curvature, from a visual/geometric viewpoint... One formulation I got was that Intrinsic ...
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1answer
71 views

Riemannian metric and geodesic

For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$. I am not sure about the ...