(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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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) ...
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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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21 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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36 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 ...
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38 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 ...
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54 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 $y_1(x_1,x_2)$, ...
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1answer
22 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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37 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 ...
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51 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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45 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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33 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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62 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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51 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 ...
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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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46 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 ...
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20 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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45 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 ...
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45 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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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
95 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 ...
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54 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 ...
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30 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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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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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 ...
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33 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 ...
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1answer
71 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 ...
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1answer
90 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 ...
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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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62 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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12 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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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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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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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
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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 ...
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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
143 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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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 ...
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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
75 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. ...
2
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1answer
78 views

Reference request-What is the prerequisite of S.S.Chern's proof of the generalised Gauss-Bonnet theorem?

The title basically explains everything. The OP is an independent learner, who in the current stage sets S.S.Chern's proof of the generalised Gauss-Bonnet theorem as the goal. But what is the ...
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1answer
69 views

Intuition/visualization for a non-flat connection

I'd just like to check whether my visualization for a way to get a non-flat connection is correct. The definition I am using for a connection is, for a fiber bundle $\rho:E \to B$, a smooth ...
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1answer
36 views

For Green's function of $\Delta-c$, how to show $\int_{X}G(x,y)(\Delta-c)f(y) dy=\pm f(x)$?

Let $X$ be a compact Riemnnian manifold and $\Delta$ the Laplacian. Suppose that $G(x,y)$ be the Green's function of the elliptic operator $\Delta-c$ for a positive constant $c$. I think the ...
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Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$, i.e., each ...
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If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$

I'm having the same problem as it was questioned here. I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n ...
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1answer
32 views

Question on Normal Coordinates

I'm having a hard time trying to understand something that I'm suspicious is pretty stupid. I'll refer to Wikipedia to settle the term's I'll refer to. ...
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1answer
64 views

Multi Index Dirac delta function

If we assume the following result: $$\delta^{\alpha_1,\alpha_2,\cdots , \alpha_k, \rho}_{\beta_1,\beta_2,\cdots , \beta_k, \rho} = (n-k)\delta^{\alpha_1,\alpha_2,\cdots , ...
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65 views

Why the geodesic curvature is invariant under isometric transformations?

As I know the geodesic curvature $$ \kappa_g = \sqrt{det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ ...
5
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1answer
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Explicit description of flat connections under pullback on principal bundles over Riemann surfaces

I'm trying to find a proof/reference for a statement that I've seen quoted in some way or the other, but without reference. The setting: let $P\longrightarrow M$ be a flat principal $G$-bundle over ...
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9 views

Short-time representation of variations of metrics on principal bundles via exp?

Let us consider a principal $G$-bundle $P\longrightarrow M$ together with an $H$-reduction $s$, where $H$ is a maximally compact Lie subgroup. As an $H$-reduction, $s\in\Gamma(M,P/H)$, hence we can ...
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Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...