(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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Showing the right half of the unit hyperbola is a complete metric space.

Let $f : \mathbb{R} \rightarrow \mathbb{R}^2$ be given as follows. $$f(\theta) = (\cosh \theta, \sinh \theta)$$ I want to argue that $\mathrm{im}(f)$ is a complete metric space with respect to the ...
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16 views

Metric for subset of $\mathbb{R}^n$

Consider the collection of points in $\mathbb{R}^n$ whose coordinates are all strictly positive. I want to think of this subset as a Riemannian manifold. Does anyone happen to know what the Reimannian ...
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23 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
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15 views

Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
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16 views

Flat connections on a manifold

I'd like to show that: For a flat connection on a manifold $M$, there is a parallel local frame near every point $p$ in $M$. I thought about using Cartan's structure equation but I can't see the end ...
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26 views

If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
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47 views

Riemann curvature product metric

Suppose that $M=M_1 \times M_2,$ with the product metric $g= g_1 \oplus g_2.$ Let $p\in M$ and suppose that $X \in T_pM_1$ and $Y\in T_pM_2.$ I want to show that $R(X,Y,Y,X)=0,$ at the point $p.$ I ...
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54 views

Topologies in a Riemannian Manifold

I'm studying Differential Manifolds using Manfredo do Carmo's Book (Riemannian Geometry) and although I see no mention of this in Do Carmo's book, it's really easy to see a Riemannian Manifold as a ...
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+100

How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
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39 views

A simple metric question

In their article Killing vector fields of standard static spacetimes, Dobarro and Unal derived the following simple identity. Note that if $h:I→R$ is smooth and $Y,Z∈ {\frak{X}}\left(I\right)$, ...
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37 views

Existence of normal orthogonal frame on sphere such that it is normal at every point in a neighbourhood?

It seems that on sphere $S^{n-1}$, there exists a better frame than the usually normal frame. In the literature, some author asserts that there exists a local orthogonal frame ...
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35 views

The uniqueness of the Einstein metric on $\mathbb CP^n$

Is the Fubini-Study metric the unique Einstein metric (up to scaling by a constant) on $\mathbb CP^n$? More restrictively, Is the Fubini-Study metric the unique Kahler-Einstein metric (up to scaling ...
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35 views

Prove that there are no complete regular minimal surfaces lying above a paraboloid

Prove that there are no complete regular minimal surfaces lying above a paraboloid contained in $U=\{(x,y,z) \in \mathbb{R}^3 : a(x^2+y^2)<z\}$. Here $a>0$. I've had this problem on my mind ...
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32 views

Can you define a projection with out inner product

Does a projection require a concept of inner-product?
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64 views

(locally) “almost convex” property of the distance function in a general Riemannian manifold

Given two constant-speed geodesics $\gamma_1$ and $\gamma_2$ in an euclidean space $\mathbb E^n$, it is possibile to see that: $$ t \mapsto d(\gamma_1(t), \gamma_2(t)) $$ is a convex function. The ...
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21 views

About Geodesic polar coordinate

What is different from geodesic polar coordinates and other polar coordinates? Geodesic polar coordinates has a form of $$ds^2=dr^2+f(r,\theta)^2\,\,d\theta^2$$ In $S^2$, $f(r,\theta)=\sin(r)$ which ...
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1answer
43 views

Question about complete metric on manifolds

I've recently been wondering about whether non-complete metrics on manifolds can be transformed into complete metrics on manifolds and whether all manifolds have complete metrics. After some googling ...
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27 views

Ricci scalar algebra

Is this derivation correct? If $R=0$, $R$ being the scalar curvature, then: $$R_{;k}=0$$ $$(g^{ac}g^{bd}R_{abcd})_{;k}=0$$ $$g^{ac}g^{bd}(R_{abcd})_{;k}=0$$ $$R_{abcd;k}=0$$
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25 views

No Lorentzian metric on $S^{2}$

In the book Riemannian Geometry by Gallot et al there is a remark at the beginning that there is no Lorentzian metric on $S^{2}$. Is it a difficult theorem? Or there is an easy solution? Any hint/idea ...
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1answer
65 views

Exponential map and distance on a Riemannian Manifold

I'm trying to solve an exercise which I thought at first seemed simple but I'm having some trouble pegging down the error term. The question is to prove that on a Riemannian manifold we have ...
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1answer
35 views

Conformal classes and almost-complex structures

It is well-known that on closed oriented surfaces $S$, conformal classes of metrics on $S$ correspond bijectively to complex structures on $S$. My understanding is that this correspondance goes as ...
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3answers
104 views

Differentiability of the distance function

Suppose that $d:X \times X \to \mathbb{R}$ is a geodesic distance function on a smooth Riemannian manifold $X$ ($d$ is determined by metric tensor) and $x \in X$ is fixed. What can be said about the ...
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36 views

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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37 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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60 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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45 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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1answer
40 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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41 views

The curvature of surfaces in Euclidean space (Theorema Egregium)

The below animation is from Wikipedia. It shows how a helicoid can be deformed into a catanoid and vice versa without stretching. Because of this, the Theorema Egregium shows that the Gaussian ...
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80 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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1answer
32 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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28 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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68 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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59 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
38 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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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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58 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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46 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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41 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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2answers
169 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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1answer
51 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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1answer
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
58 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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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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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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49 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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29 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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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 ...