Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

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13 views

Proof that $D_XY=(X^i\partial_i Y^j) \partial_j$ is metric connection on $\mathbb{R}^3$

I need to proof $D_XY=(X^i\partial_i Y^j) \partial_j$ is metric connection on $\mathbb{R}^3$ I want to use metric tensor (dot product). So I must show that $Z(X \cdot Y)=D_ZX \cdot Y +X \cdot D_ZY$. I ...
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10 views

Surface gradient of a vector field

Let $\mathbb{M} \subset \mathbb{R}^3$ be a $C^2$ manifold with normal vector ${\bf n}$. Then, for any scalar-valued function $f$, we can write its surface gradient as: \begin{align} \nabla_{\mathbb{M}}...
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1answer
34 views

When can the trajectories of a vector field be realized as geodesics?

Suppose we have a smooth manifold $M$ and a vector field $X$ on $M$. Then the trajectories of $X$ are pairwise non-intersecting, and further, each trajectory is either injective or periodic. I'm ...
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39 views

The Ricci tensor is independent of the coordinates.

Could you help me with the following problem, please? What I will write can be found in the book Introduction to Riemannian geometry with applications to mechanics and relativity by Leonor Godinho and ...
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25 views

Changing the Riemannian metric does not affect the $Spin^c$-structure

It is well known that changing the Riemannian metric on a manifold does not change the $Spin$-structure. I suppose the same should be true for $Spin^c$-structures, but I am unable to prove it or to ...
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57 views

Spectrum of $\Delta+a$ on a compact manifold, where $a$ is a function

Let $(M,g)$ be a closed Riemannian manifold and $\Delta$ be its Laplace-Beltrami operator. It is well known that the spectrum of $\Delta$ is a discrete subset of $[0,+\infty)$ and the eigenfunctions ...
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1answer
28 views

Show that the angle between $X$ and $Y$ is the same under the standard and hyperbolic metrics.

Consider the following two metrics on $\mathbb{H}=\{(x^1,x^2)\in \mathbb{R^2} \ | \ x^2>0\}$: $$g_0=dx^1\otimes dx^1+dx^2\otimes dx^2, \ \text{(standard)}$$ $$g_1=\frac{1}{(x^2)^2} dx^1\otimes dx^1+...
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45 views

Riemannian textbook reference request on umbilic points and totally umbilic hypersurfaces

I'm doing a project exploring Riemannian immersions and submersions. In O'Neil's book, Semi-Riemannian Geometry, he mentions umbilic points and totally umbilic hypersurfaces and proves some ...
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1answer
38 views

Notation confusion in time-derivative: $g_t(X,X)=g_t^{ij}X_jX_i=\sum X_i^2$

Let $(M,g)$ a Riemannian manifold and $X$ a vector field. I know that $g(X,X)=g^{ij}X_jX_i=X^iX_i$. Some authors use $g(X,X)=\sum X_i^2$ instead. That was Ok until I wanted to compute the time-...
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23 views

Betti number of sphere and vector-Laplacian

I am a physics student, and I am having trouble understanding the relationship between Laplacian and Euler numbers through concrete calculations. In my understanding, the "dimension" of the ...
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1answer
46 views

Volume form of manifold with metric $ds^2=\frac{dx^2}{x^2}+\frac{dy^2}{y^2}$

How do you write $ds^2=\frac{dx^2}{x^2}+\frac{dy^2}{y^2}$ in matrix form? Is it $\frac{1}{x^2}$ and $\frac{1}{y^2}$ on the diagonal and $0$ else-where? I need this to calculate the volume form. $g=\...
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42 views

Gauss Bonnet Theorem for half a cone

Let $S \subset \mathbb{R}^3$ be given by $$ S = \{ (x,y,z) \in \mathbb{R}^3 \, : \, x^2+y^2=z^2 , \quad y \ge0, 0\le z \le 1 \}. $$ Verify the Gauss Bonnet Theorem, by computing $\int_S K\, dA$ and $\...
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28 views

Every Riemannian metric is a pseudo-Riemannian metric

Let me introduce some definitions first in order for our discussion. A Riemannian metric on a smooth manifold $M$ is a smooth symmetric $2$-tensor field $g$ on $M$ that is positive-definite at each ...
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27 views

Product manifold volume element

Let $M=N_1 \times N_2$ be a 4-dimensional Riemannian product manifold with metric $g_M=g_{N_1} + g_{N_2}$, where $g_{N_1}=g_{ij}du^i du^j$ (with $i, j= 1, 2$) and $g_{N_2}=g_{ab}dv^a dv^b$ (with $a, b=...
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55 views

The geodesic differential equations in an orthogonal coordinate system

Prove that for an orthogonal coordinate system the geodesic differential equations become $$\frac{d}{ds}[g_{ii}(\frac{dx^i}{ds})]=\frac{1}{2}\sum_{j=1}^{n}\frac{\partial g_{jj}}{\partial x^i}(\frac{dx^...
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24 views

Regular embedded manifold, isometric embedded

In "On Einsteins Path" Chapter 11 "Wave Maps in General Relativity", I have seen the following definition for a "regularly embedded manifold $(M,g)$: Let $q$ be a Euclidean ...
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46 views

The hyperbolic plane $\mathbb{H}^2$ can't be isometrically immersed in $\mathbb{R}^3$

It's easy to note that there is a local isometry between the hyperbolic plane $\mathbb{H}^2$ and the pseudosphere, since they have constant curvature equal to $-1$. Hilbert's Theorem.- There exists no ...
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1answer
46 views

Example of Riemannian metric on sphere such that it become non strictly convex.

My understanding of strictly convexity of the compact set in Euclidean space is that if we take any straight line joining any two boundary points then the line must be in a compact set with out ...
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1answer
30 views

Mean curvature equals to the Laplacian of the position vector?

My question is based on the manuscript titled On Mean Curvature Diffusion in Nonlinear Image Filtering (link: https://escholarship.org/content/qt0736r63m/...
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51 views

What is the analogous of $\|\nabla X\|^2=\sum_{i=1}^ng(\nabla_i X,\nabla_i X)$ for two forms?

For a vector field $X$ it is well-known that $\|\nabla X\|^2=\langle\nabla X,\nabla X\rangle=\sum_{i=1}^ng(\nabla_{e_i} X,\nabla_{e_i} X)$ where $\{e_i\}$ is an ONB and $\langle \cdot,\cdot \rangle$ ...
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54 views

Regularly embedded manifold

In "On Einsteins Path" Chapter 11 "Wave Maps in General Relativity", I have seen the following definition for a "regularly embedded manifold $(M,g)$: Let $q$ be a Euclidean ...
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1answer
33 views

Compact manifold with smooth boundary embedded isometrically into compact manifold without boundary with same dimension as original one.

Let $(M, g)$ be a compact manifold with a smooth boundary. There is a compact connected manifold without boundary $(N, g)$ having the same dimension as $M$ so that $(M, g)$ is isometrically embedded ...
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36 views

Example of compact manifold with bdd such that there exists geodesic whose first time of hitting boundary is different form exit time of geodesic,

Let M be compact Riemann manifold with boundary. The unit sphere bundle $SM$ is given by $$SM=\{(x, v)||v|_{g}=1,x \in M\}$$ where $g$ is the Riemannian metric in the tangent space at $x$. Given $(x, ...
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1answer
36 views

Bounding integrals on a `moving' domain

I've been reading through Schoen-Yau's proof of the positive mass theorem and find myself stuck on a particular estimate they prove. It seems to be simple multivariable calculus, but I haven't been ...
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1answer
84 views

Hairy Ball Theorem on $\mathbb{S}^2$: a counter-example?

Let $(U, (\theta, \varphi))$ be the spherical coordinate chart on the sphere $\mathbb{S}^2$, and consider the vector field on $U$ defined by $\dfrac{\partial}{\partial \varphi}$. Then in this chart, ...
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1answer
46 views

Example of manifold with geodesic having infinite exit time and finite negative exit time.

Let M be compact Riemann manifold. The unit sphere bundle $SM$ is given by $$SM=\{(x, v)||v|_{g}=1,x \in M\}$$ where $g$ is the Riemannian metric in the tangent space at $x$. Given $(x, v) \in S M$, ...
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74 views

What is the minimal length of a “Diagonal” in a Torus?

Given a Riemannian torus $(T,d)$ with fundamental group $\pi_1(T)=\langle a,b \mid ab=ba \rangle$. Denote for any $\gamma \in \pi_1(T)$ the infimum length of all representatives of $\gamma$ by $L(\...
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55 views

Proof of $\int_Mf\nabla_i\nabla_jh_{ij}\mathsf{dvol}_g=\int_M(\nabla_i\nabla_j f)h_{ij}\ \mathsf{dvol}_g.$

Suppose $f$ is a scalar function and $h$ a symmetric $(0,2)$ tensor on a closed Riemannian manifold $(M,g)$. Then is the following equality true? $$\int_Mf\nabla_i\nabla_jh_{ij}\mathsf{dvol}_g=\int_M(\...
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42 views

Cylinder Parallel Transport

In general the parallel transport of a certain vector between two points p and q in a given surface may change depending on the curve used to go from p to q. However, in a cylinder the parallel ...
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1answer
31 views

Error-term in the expansion of a riemannian metric in normal coordinates

Let $(M,g)$ be a Riemannian metric and $\exp_x^{-1}:B_r(x) \to B_r(0) \subset \mathbb{R}^n$ be a normal coordinate system around $x \in M$ with respect to some orthonormal basis $e_1,\ldots, e_n$. The ...
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43 views

Show that two Riemannian manifolds are isometric.

Let $\mathbb B^n(R)$ the ball of radius $R$ centered at the origin in $\mathbb R^n$, with the metric given in coordinates $(u_1,\cdots,u_n)$ by : $$h_B=\dfrac{4R^4}{(R^2-|u|^2)^2}\sum_{i=1}^n(du_i)^2$$...
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1answer
24 views

Injectivity radius and complete manifold

Let $(M,g)$ be a Riemannian manifold. $M$ is called complete, if the maximal geodesics on $M$ are defined on all of $\mathbb{R}$. The injectivity radius of a point in $M$ is the largest radius, s.t. ...
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Let A, B be nowhere zero smooth functions on $\mathbb{R}^2$ and consider the Riemannian metric $g =A^2 dx^2 + B^2 dy^2$.

Let A, B be nowhere zero smooth functions on $\mathbb{R}^2$ and consider the Riemannian metric $g =A^2 dx^2 + B^2 dy^2$, where $x, y$ are the standard coordinates on $\mathbb{R}^2$. a. Compute the ...
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23 views

Prove that the scalar curvature is twice the value of the Gaussian curvature

I would like to show that the scalar curvature $R$ is twice the value of the Gaussian curvature $K$: $$R=2K.$$ I know that $$R=g^{ij}R_{ij}=Kg^{ij}g_{ij}.$$ But somehow I don't know what's going on in ...
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41 views

Why is the covariant derivative defined as $\nabla_X Y = P((X\otimes I)Y)$ in the Wikipedia article “Riemannian connection on a surface”?

In the Wikipedia article Riemannian connection on a surface, I am struggling to interpret the following: "Since $\mathcal X(M)$ is a submodule of $C^\infty(M,\mathbf E^3)=C^\infty(M)\otimes \...
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41 views

The stereographic projection is a conformal transformation.

The stereographic projection $\sigma :\mathbb S^n(R) \setminus\{N\} \to \mathbb R^n$ is a conformal transformation . To prove this theorem, we should show that for all $v\in T_q\mathbb R^n$ ($q\in \...
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41 views

Solution to the wave equation on a Lorentzian manifold

I want to prove global existence of a wave map on a Robertson Walker spacetime $V= I \times S$, where $I=[0, \infty) \subset \mathbb{R}$ with metric $$ g=-dt^2+R^2 \sigma $$ where $\sigma$ is a ...
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110 views

Is every compact $n$-manifold that immerses in $\mathbb{R}^{n+1}$ smoothable?

Suppose that $M$ is a compact topological $n$-manifold, and $f: M \rightarrow \mathbb{R}^{n+1}$ is a topological immersion, i.e. a local topological embedding. Then is $M$ smoothable? By that, I ...
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Hessian of a vector field

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$. For a scalar function $f$, one has the Hessian formula \begin{equation*} \nabla^2f(X,Y) = \nabla_{X}\nabla_{Y}f - \nabla_{\...
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56 views

Zero Locus of Killing Vector Field $X$ and Rank of Differential Form $dX$

Say $X$ is a Killing Vector Field on an $n$-dimensional Riemannian Manifold. We know that each connected component of the zero set is a totally geodesic submanifold of even codimension. In the ...
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25 views

The existence of right invariant metric on the left-coset homogeneous space $G/\Gamma$ [closed]

Let $G$ be a Lie group and $\Gamma$ be a lattice in $G$. Consider the $G$-homogeneous space $G/\Gamma$, I frequently see from research papers that $G/\Gamma$ admits a right-invariant metric but I don'...
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42 views

How to check if the following is a Green's function

Let $(M,g)$ be a four dimensional lorentzian manifold and $M\overset{\rho}{\longrightarrow}\mathbb{R}$ be a smooth function. I am trying to solve the following wave equation. $$\nabla_\mu\partial^\mu \...
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1answer
40 views

Globally hyperbolic Lorentzian manifold

I am currently trying to work through properties of globally hyperbolic Lorentzian manifolds and there are some things which aren't clear to me: I have the following definition of globally hyperbolic: ...
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1answer
58 views

Extrinsic definition of a Riemannian manifold?

In every book that I have read (Bott and Tu, Absil), Manifolds are defined intrinsically in terms of charts and maximal atlases, after which we show how to "construct" manifolds from level ...
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2answers
45 views

Finite volume conditions of manifolds

I was reading this question and wondering if there are more general results answering the question "Under which condition do manifolds have to have a finite riemannian volume?". I know that ...
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1answer
35 views

Geodesic orthocenter

Suppose $(M,g)$ is a complete Riemann manifold. Is it true that for any $A,B,C \in M$ that do not lie on the same geodesic, there exists an $H \in M$ such that $A,B,C,H$ lie in a totally geodesic ...
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1answer
39 views

In what sense is the Ricci Curvature Tensor an “average”?

I’ve heard that the Ricci tensor loosely represents an “average” curvature of some sort. The physical meaning of the Riemann tensor is relatively clear to me as I’ve seen how to derive it, but the ...
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2answers
59 views

Geodesic completeness of manifolds with “warped” metrics

Let $(M,g)$ be a geodesically complete Riemannian manifold, and let $f: M\to \mathbb{R}^+$ be a smooth, bounded strictly positive function on $M$; i.e., there exist $L,A\in \mathbb{R}^+$ such that $0&...
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1answer
37 views

$C^1$ manifold curve with bounded derivative has limit

Let $\mathcal{M}$ be a smooth Riemannian manifold (not necessarily complete) and consider $\gamma : [0,1) \to \mathcal{M}$ a differentiable curve with bounded derivative, i.e., there exists constant $...
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13 views

Relationship between the $\mathtt{Exp}^{-1}$ mapping of three points on a common geodesic on a Riemann manifold

Suppose that there is a Riemann manifold $\mathcal{M}$ and $x,z$ be two point on $\mathcal{M}$. Let \begin{equation} y = \mathtt{Exp}_{x}\left( t \cdot \mathtt{Exp}_{x}^{-1} \left( z \right)\right) \...

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