(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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21 views

What is the exterior normal to the boundary of a Riemannian manifold?

Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $p \in \partial M$ and in local coordinates $(x_1,\ldots,x_n)$ near $p = (0,\ldots,0)$ the manifold $M$ is given by $\{x_n ...
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19 views

Question about a particular estimate in Riemannian geometry.

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
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1answer
24 views

Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$?

Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$ ...
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0answers
25 views

Reference for construction on Riemannian Manifolds

I would like to know a book or an article where the connected sum of Riemannian manifolds is explained with some detail. I roughly know how to do the construction but I want to be able to check a more ...
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0answers
47 views

Riemannian Submerssion

I am reading John Lee's Riemannian Geometry Chapter 3, and I want to do some exercises. I think that I need some hints to solve the following: (Problem 3-8 of that book) Suppose $M$ and $N$ are ...
4
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1answer
52 views

Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$ TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-)) $$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
2
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0answers
19 views

Generalizations of the product neighborhood theorem

As far as I know, the Product Neighborhood Theorem for smooth manifolds states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization ...
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0answers
15 views

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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3 views

Perturbation of the boundary of a strictly stable minimal surface

Let $\Sigma \subseteq \mathbb{R}^3$ be a minimal surface with boundary $\Gamma$. Now let us assume that $\Sigma$ is strictly stable, that is, $\lambda_1(\Sigma,L) >0$, where $L$ is the stability ...
7
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1answer
79 views

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 ...
0
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1answer
42 views

What are the laplacian operators for the three two dimensional metrics of one variable dependence?

What are the laplacian operators from the three following two dimensional metrics of one variable dependence : \begin{align} (A) && d\mathcal{l}^2 = e^{2w(x_2)} \left( dx_1^2 + dx_2^2 \right) ...
4
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2answers
72 views

Known methods to detect coordinate singularities?

MOTIVATION In Riemannian geometry, when one writes the metric tensor in a particular coordinate system, certain 'fake' singularities might appear that have little to do with the geometry near them. ...
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0answers
17 views

Reference frame rotation depending on metric tensor $g_{\mu\nu}$

My simple question is: The infinitesimal line element is $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$$ where $g_{\mu\nu}$ is the metric tensor of the space. Is it possible from the simple knowledge of ...
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0answers
34 views

Is this cohomology isomorphic to De Rham Cohomology?

Let $(M,g)$ be a Riemannian manifold. Put $d^{*}= *d*$ where $*$ is the Hodge $*$ operator. So $d^{*}\circ d^{*}=0$. Then it introduce a (c0)homology. What is a relation between this ...
0
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1answer
35 views

Hyperbolic metric of arbitrary curvature.

I've been trying to find this online, in books, etc, but I can never find the expression for the metric on the unit disk $$\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$$ that has constant ...
0
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1answer
51 views

How to solve for the extrinsic variables of a one variable scaled conformal metric to an equivalent metric?

Given the following metric equivalence \begin{align} e^{2w(x_2)} \left( dx_1^2+dx_2^2 \right) = dy_1^2+dy_2^2+dy_3^2 \end{align} is their a known solution for the extrinsic variables $y_1(x_1,x_2)$, ...
2
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0answers
12 views

Local geodesics in uniquely geodesic spaces

Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a ...
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0answers
35 views

Scwarzschild solution question

Since we set the ricci tensor to be zero everywhere, why is it still a solution if it doesn't apply to the point where the point mass exists? Shouldn't it apply also to that point as well, or am I ...
1
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2answers
47 views

Question about a notation. Norm of the derivative of a function at a point

Given is an analytic function from $M$ to $N$, both equipped with conformal Riemannian metric, say $g$ and $h$ resp. What might the $h$ norm of the derivative of the function at a point mean? ...
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1answer
37 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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0answers
32 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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0answers
58 views

Calculate Geodesic Path of $N\times N$ matrix on Riemannian manifold of fixed rank

If I have two matrices $A(0)$ at $t=0$ and $A(1)$ at $t=1$, they are $N\times N$ matrices, and they are on the Riemannian manifold of rank $K$. How to calculate the geodesic path $A(t)$? I haven't ...
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0answers
48 views

Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
8
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1answer
126 views

Is every scalar differential operator on $(M,g)$ that commutes with isometries a polynomial of the Laplacian?

On $(\mathbb{R}^n, g_{\text{std}})$ with $\Delta$ the Laplacian, the following holds: Fact: Every scalar differential operator $D$ that satisfies $D \circ F^* = F^* \circ D$ for all isometries $F ...
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1answer
41 views

Calculate geodesic path on matrix manifold

I have a matrix which is change with time. Let me denote it as A(t). I know t=0 it is A(0) and I know t=1 it is A(1). A is symmetric positive semi-definite matrix. What I want to do is find the ...
0
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1answer
18 views

Boundedness of Riemannian curvature gradient

I'm reading a paper by Wan-Xiong Shi "Deforming the metric on complete Riemannian manifolds". And there is a statement without proof. It can be summarized as follows: Let $B(x_{0},\gamma)$ be a ...
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42 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
2
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1answer
42 views

Is a connected compact Riemannian manifold of dimension 1 unique?

The tiles says almost everything. It is known that a connected compact topological manifold of dimension 1 is isomorphic to $S^1$. What if we replace "topological" by "riemannian"?
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20 views

Reformating Function

Is there such a function where a ambiguous ;n-dimensional, field/space (defined by a function) is plugged in and returns a flattened field where the basic units along the function are then formatted ...
3
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1answer
86 views

Connections in non-Riemannian geometry

In case of Riemannian geometry the connection $\Gamma^i_{jk}$ as is derived from the derivatives of the metric tensor $g_{ij}$ is ought to be symmetric wrt to its lower two indices. But in the case of ...
2
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1answer
50 views

Extension of smooth maps at a cusp

There is a short remark in deCarmos "Riemannian Geometry" (p. 67) and I wonder about the condition that the vertex angles must be $\neq \pi$. If $s_1$ and $s_2$ are two differentiable maps on an ...
0
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1answer
28 views

Do we need a metric to define plurisubharmonic functions?

There are various notions of 'harmonicity' on various manifold. Sometimes, I am counfuesed by the definitions. For real manifold, the harmonic manifold is defined by $\Delta f=0$, where $\Delta$ is ...
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0answers
53 views

Special expression for 3-linear symmetric map $T(X,Y,Z) = \langle -Jh(X,Y),Z \rangle$ [Ejiri]

For a project in Riemannian Geometry, I have been working out the details of a paper by Ejiri (http://www.ams.org/journals/proc/1982-084-02/S0002-9939-1982-0637177-8/S0002-9939-1982-0637177-8.pdf) ...
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0answers
30 views

Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
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0answers
52 views

All differentiable functions on $\mathbb{S}^n$

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
1
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1answer
30 views

Hyperbolic (and related) structures on open unit disk

I am facing some confusion about different structures on the open unit disk $D:=\{ z \in \mathbb{C}, |z|<1 \}$. By Riemann Mapping Theorem we know there is just one complex structure on $D$, up to ...
1
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1answer
66 views

Reference about Gauss-Bonnet-Chern theorem.

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
2
votes
1answer
86 views

When is the Hessian contracted with a vector field a closed form?

Suppose M is a Riemannian manifold and $g, f : M \rightarrow \mathbb{R}$ are smooth functions. When is the $1$-form Hess$^f(\nabla g, -)$ closed? I'm looking for simple conditions involving f,g and ...
3
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2answers
91 views

About two notions of holonomy

I have found something called "holonomy" in two apparently different contexts: Let $M$ be a smooth manifold, $E\to M$ a vector bundle and $\nabla $ a connection on $E$. Then you have a notion of ...
4
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2answers
60 views

Orientation on Riemann surfaces

$\mathcal{X}$ is a Riemann surface and $\mathcal{E}^{(2)}(\mathcal{X})$ is the $\mathbb{C}$-Vector space of all differentiable $2$-forms on $\mathcal{X}$. I want to define the orientation of ...
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0answers
10 views

Existence of biregular cover on a Riemannian manifold with a codimension one foliation.

Suppose we have a (psaudo-)Riemannian manifold $(M,g)$ of dimension $m$ and that a codimension one foliation $\mathcal{F}$ exist for $M$. By Frobenius theorem this foliation is in one-to-one ...
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0answers
34 views

Volume of the ball in smooth Riemannian manifold

If V(x,r) is the volume of a ball B(x, r) on a smooth Riemannian manifold, for fixed point x, is the function V(x, r) continue or differential for radius r?
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1answer
43 views

Length Minimizing Properties of Geodesics on Surfaces?

Can anyone recommend me some nice references about lengh minimizing properties of geodesics? I'm looking for a treatment in the case of surfaces, but more general viewpoints will also be welcome. ...
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0answers
22 views

There are no conjugate points on a surface with negative Gaussian curvature?

I'm trying to understand the following theorem about conjugate points: Theorem. Let $M$ be a complete surface with Gaussian curvature $K\leq 0$, then there are no conjugate points on $M$. Proof: Let ...
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1answer
63 views

Can I construct an affine connection on a Riemannian manifold from arbitrary Christoffel Symbols?

The question is rather simple. All my definitions are as in Do Carmo's "Riemannian Geometry". If $M$ is a Riemannian Manifold, can I construct an affine connection $\nabla$ on it by setting, for all ...
2
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1answer
45 views

Computing the volume element of an oriented Riemannian manifold

I'm reading Gallot-Hulin-Lafontaine, and in section 2.7 they say they following: I wanted to check that the second $v_g,$ given in a local oriented chart, satisfied the first property. So I ...
2
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1answer
140 views

Why do these geometric assumptions imply these statements about relative homology?

I'm reading the paper Coverage in sensor networks via persistent homology. As in the paper, let $\mathcal{D}$ be a bounded domain in $\mathbf{R}^d$. We make the following assumptions: A5 The ...
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0answers
29 views

Isometries and geodesics in projective plane using covering

We define a relation in the sphere by identifying the antipodal points, the quotient space obtained is the projective plane $\mathbb{P}^2$. Also, the quotient map ...
0
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1answer
39 views

Derivative of riemannian metric

I dont understand the following detail $$ \frac{1}{2} \int_a^b \frac{d}{dt}(g(X,X)ds = \int_a^bg(\nabla_YX, X)$$ Here $X = d\phi (\partial/\partial s)$ and $Y =d\phi (\partial/\partial t)$. Where ...
2
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2answers
70 views

Ricci Soliton geometric meaning

I wonder what is the geometrical, intuitive meaning of a Ricci soliton on a manifold. The definition that I use is as follows. $V$ is a vector field on the manifold, $g$ is a Riemannian metric. ...