(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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The finiteness of the fundamental group of a closed Ricci-flat manifold

If $M$ is a Ricci-flat closed Riemannian manifold with $H^1(M,\mathbb R^n)=0$, can we show the fundamental group of $M$ is finite?
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39 views

The isometry group of the simply-connected Ricci-flat closed manifold

If $M$ is a simply-connected Ricci-flat closed manifold, then is $I(M)$ the isometry group finite?
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75 views

Distance between two points on the Clifford torus

How can I obtain the distance between two points $\mathbf{x}=(x_1,x_2,x_3,x_4)$ and $\mathbf{y}=(y_1,y_2,y_3,y_4)$ that belong to the $2$-torus $\mathbb{S}^1\times \mathbb{S}^1$? This is, I want to ...
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51 views

Ricci SCALAR curvature

Are there any manifolds that have NULL SCALAR curvature but not null ricci curvature tensor? For dim>2, obviously. Edit: Are there, also, any manifolds of null scalar curvature but ricci curvature ...
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43 views

A question on self-dual differential 2-forms

This question is from Lemma 2 in Derdzinski's paper. Let $$\omega=e_1\wedge e_2+e_3\wedge e_4, \eta=e_1\wedge e_3+e_4\wedge e_2, \theta=e_1\wedge e_4+e_2\wedge e_3$$ be a basis for self-dual ...
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95 views

Pushforward along an exponential map

My differential geometry is a bit rusty, so I'd like some help with what follows: I have the following setting: $M,N$ Riemannian manifold of dimension $m<n$ with codimension $d$, $M$ in embedded ...
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51 views

Is there an efficient way to prove orthogonality of a coordinate system?

Suppose we define a new orthogonal coordinate system, such as spherical coordinates defined by $$x = r \sin \theta \cos \phi, y = r \sin \theta \sin \phi, z = r \cos \theta.$$ Is there an efficient ...
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29 views

Version of gauss theorem

I came across a statement which says that the version of Gauss-Theorem says that : $$\int_{\partial \Omega } f div_{\partial \Omega } v ds = - \int_{\partial \Omega} v \cdot \nabla^{\tau} f ds + ...
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73 views

Is there an easy way to reason about expressions involving lots of indices?

I have been reading some Riemannian geometry recently. So far, I think I am understanding the concepts well enough. However, I am finding it difficult to translate some of the notation into meaning. ...
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80 views

On local isometry of mappings

Apropos discussion at this link: locally isometric is not a symmetric relation. My instructor told me that a counterexample to the symmetric relation here could be actually even simpler than the ...
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1answer
48 views

The formula for a distance between two point on Riemannian manifold

Given a Riemannian (say, connected) manifold $(M,g)$ one can produce the metric via a formula $d(x,y)=\inf_{\gamma}l(\gamma)$ where $\gamma$ is piecewise smooth curve joining $x$ and $y$. My question ...
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138 views

Is there a codifferential for a covariant exterior derivative?

For forms on a Riemannian $n$-manifold $(M,g)$ there is a notion of a codifferential $\delta$, which is adjoint to the exterior derivative: $$\int \langle d \alpha, \beta \rangle \operatorname{vol} = ...
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68 views

Naturality of the pullback connection

I'm completely stuck proving the naturality of the pullback connection. The strategy suggested is a follows: We let $\phi: (M,g) \to (\tilde{M}, \tilde{g})$ be an isometry, with connections ...
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48 views

Noncomplete riemannian manifolds

In Lee's Riemannian manifolds text, he claims that "on $\mathbb{R}^n$ with metric $(\sigma^{-1})^* g$ obtained from the sphere from stereographic projection, there are geodesics that escape to ...
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15 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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57 views

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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175 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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13 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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55 views

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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60 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
63 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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56 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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40 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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52 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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30 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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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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61 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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115 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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34 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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35 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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23 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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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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41 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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79 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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31 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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49 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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33 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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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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27 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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33 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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48 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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56 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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106 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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43 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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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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42 views

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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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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32 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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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 ...