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Questions tagged [riemannian-geometry]

A branch of differential geometry dealing with Riemannian manifolds. Riemannian manifolds are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag if ...

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Prove that the paraboloid in $R^3$, defined by $x^2 + y^2 - z^2 =a$ is a manifold if $a>0$. why does not $x^2 + y^2 -z^2 =0$ define a manifold?

Prove that the paraboloid in $R^3$, defined by $x^2 + y^2 - z^2 =a$ is a manifold if $a>0$. why does not $x^2 + y^2 -z^2 =0$ define a manifold? Could anyone give me a hint of the solution of this ...
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21 views

Existence of boundary cylindrical neighborhood for a compact manifold

Let $M$ be a compact riemannian manifold with boundary $\partial M\neq \varnothing$. I would like to show that there is some neighborhood $U$ of $\partial M$ which is diffeomorphic to $[0,a)\times \...
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20 views

Exhaustion in compact convex (totally geodesic) subsets of a complete manifold

Consider a complete Riemannian manifold $M$. We know that it admits an exhaustion in compact subsets. My question is: can I assume all these subsets to be totally geodesic, by some operation of ...
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Detail in Perelman's proof of the Soul Conjecture: why $O(\delta^2)$ and not $O(\delta)$?

Referring to G. Perelman, Proof of the soul conjecture by Cheeger and Gromoll. Given a distance-nonincreasing retraction $P$ from an open complete manifold of nonnegative curvature onto its soul $S$, ...
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8 views

Equivalent definitions of Kähler metric on a Riemannian manifold.

In the book Compact Manifolds with Special Holonomy by Dominic Joyce, Ch. 4.4 there is a proposition 4.4.2 Let $M$ be a manifold of dimension $2m$, $J$ an almost complex structure on $M$, and $g$...
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23 views

Space of divergence-free vector fields on a Riemannian manifold

Let $(M,h)$ be a smooth Riemannian manifold of dimension $d\geq 1$ with smooth metric. Set $X:=\{A= \mbox{smooth vector field s.t. } div_h A=0 \}$. Then $X$ is an infinite dimensional vector space. ...
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22 views

Reparameterized geodesics - proof

I was reading the proof for the lemma that states that a certain curve c is a reparameterized geodesic if and only if it satisfies $\frac{D\dot{c}}{dt}=f(t)\dot{c}$. The starting point was defining ...
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Proving $\left(\mathbb{R}^2_+,\frac{1}{y}(dx^2+dy^2)\right)$ is not complete

Let $\mathbb{R}^2_+:=\{(x,y)\in\mathbb{R}^2\mid y>0\}$ and the metric $g=\frac{1}{y}(dx^2+dy^2)$. Prove $(\mathbb{R}^2_+,g)$ is not complete. I guess the proper way is to find a divergent curve $\...
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+100

An isometry between totally geodesic submanifolds is smooth?

I am studying Sharafutdinov's Convex sets in a manifold on nonnegative curvature. I have found the following statement, $S_0$ being a totally geodesic submanifold of an open Riemannian variety $M$ ...
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Conformal metrics

While studying complex structures on Riemannian manifolds I found some trouble in proving this claim: "There is no metric conformal to the Euclidean one on $\mathbb{C}-{0}$ whose Gaussian curvature ...
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+50

Calibrator for unit speed curve

Non-existence of Almost Calibrator of Circle : Assume that $U$ is $\varepsilon$-open ball at origin in $\mathbb{R}^2$. Prove that there does not exist the following function : $f: U\rightarrow \...
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Determining the projections onto the Horizontal and Vertical Space

I am looking at a map $\Pi: GL(n)\mapsto Sym^+(n)$ via $\Pi(A)=AA^T$. This has differential $d\Pi_A(Z)=ZA^T+AZ^T$. The vertical space $\mathcal{V}_A$ is the kernel of $d\Pi_A$, and so $$Z\in\mathcal{...
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32 views

Finite injectivity radius implies compactness

Let $(M,g)$ be a complete Riemannian manifold. Let $\tau:SM\to\mathbb{R}$ denote the cut distance function, where $SM$ is the unit-tangent bundle of $M$. Let $i_0(p)$ denote the injectivity radius ...
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How do lie symmetries manfest in the metric tensors of their manifolds?

Suppose we are considering some pseudo-Riemannian manifold (spacetime) that has an underlying topology $SU(2)\times U(1)$ (which incidentally corresponds to a closed cyclic universe). Such a space is ...
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Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $d$-dimensional spacetime $(M,g)$ (or simply just for $d=4$): Does the conformal ...
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Does divergence theorem require smoothness?

Usually, one writes the divergence theorem as \begin{equation} \int_\mathcal{M} d^4x \sqrt{-g} \nabla_\mu v^\mu=\int_{\partial \mathcal{M}} d\Sigma_\mu v^\mu\\,. \end{equation} for some vector field $...
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37 views

(pseudo-)Riemannian manifolds and global coordinates

I have a question about (pseudo-)Riemannian manifolds. In general relativity one often assumes that there exists a global set of coordinates describing the whole manifold. My question is: Does there ...
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1answer
41 views

Finding the geodesic equations on $S^2$

The metric on $S^2$ is given as $\bar{g} = ds^2 = d\theta^2 + \sin^2\theta d\phi^2$ where $x^1 = \theta$ and $x^2 = \phi$. The only non-zero components of the Christoffel symbols are $\Gamma^{\ 1}_{ \ ...
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1answer
38 views

Normalisation of the cuspidal curve as a Riemann surface

I am trying to find the normalisation of the cuspidal curve $0=z^2 -w^3$ without its singularity, and following Simon Donaldson's book on Riemann surfaces, it seems I should find the critical points $...
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1answer
44 views

Clarifying the definition of Lie derivative for tensors

Let $(M,g)$ be a Riemannian manifold and $T$ be a smooth tensor field. The official definition of the Lie derivative of $T$ with respect to a field $X\in\Gamma(TM)$ is: $$\mathcal{L}_X(T)_p:=\left.\...
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28 views

Hyperbolic space is complete

I am trying to prove the hyperbolic space is complete. It looks one need to apply Hopf-Rinow theorem, but I don't know what to start. More precisely, I don't know what is a good way to show the ...
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1answer
39 views

Horizontal Subbundles and Connection Maps

I'm trying to see the equivalence between the Ehresmann connection and the connection map, and having trouble getting the setup to be correct. Suppose $M$ is a smooth manifold. Let $\pi:TM\to M$ ...
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Shape operators with positive eigenvalues

I am working on the frame of singular Riemannian foliations and I am interested on examples of foliations whose eigenvalues of the shape operator of a given leaf are all positive. One intermediate ...
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24 views

Laplacian on 1-forms on $(\mathbb R^m,\delta)$

I'm trying to show that if $\omega$ is a 1-form on $(\mathbb R^m,\delta)$, the action of the Laplacian is given by $$\Delta\omega=-\sum_{\mu=1}^m\frac{\partial^2\omega_\nu}{\partial x^\mu\partial x^\...
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1answer
16 views

Is there a unique minimizing geodesic to a geodesically convex set?

Let $(\mathcal M,g)$ be a geodesically complete Riemannian manifold, and $\mathcal X\subset \mathcal M$ be a compact, geodesically convex subset. It is trivial that for any point $x\in\mathcal M$, ...
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124 views

How do I prove this map is a covering Projection

Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...
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1answer
40 views

Showing that the geodesics in hyperbolic upper half-plane model are half circles

I am currently working on some differential geometry and I came across the following question. Let $H = \{(x,y)\in \mathbb{R}^2|y>0\}$, this is the hyperbolic plane (the upper half-plane model). ...
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27 views

$Ric(\nabla f ,\nabla h)$ in local coordinates, for $f,g \in C^{3}(M)$

Let $(M,g)$ a Riemannian manifold. I'm having difficulties for find the expression for $Ric(\nabla f ,\nabla h)$, where $f,g \in C^{3}(M)$. My notes says that $Ric(\nabla f ,\nabla h)=g^{ij}g^{kl}\...
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1answer
131 views

Realizing the wedge product of $\mathbb{R}^3$ as torsion or curvature tensor

Is there a connection on $\mathbb{R}^3$ whose corresponding torsion tensor is the standard cross product of $\mathbb{R}^3$, that is $T(X, Y)= X \wedge Y$? Is there a metric connection with this ...
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173 views

Transformation law for the Ricci curvature

In 4 dimensions, for a conformal change of metric $g=e^{2u}g_0$ the Ricci curvature tensor $\operatorname{Ric}$ satisfies the transformation law \begin{equation}\tag{1} \operatorname{Ric}_g = \...
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Is determining a geodesic on a riemannian manifold a limit case of the shortest path problem in a graph?

I would like to know if there is a natural way of transforming a graph in which one seeks the shortest path between two nodes into a riemannian manifold whose this graph is a subspace. My idea is to ...
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1answer
65 views

Linear functions on Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold with the corresponding $LC$ connection $\nabla$. For a smooth function $f:M \to \mathbb{R}$, the Hessian of $f$ is defined as a 2-linear map on the ...
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Relation between Hodge star operator of two related metric

Let $g,h$ be two Riemannian metric on $M$ related by $g=h+\phi$ where $\phi$ is a symmetric $(0,2)$-tensor and $g$ transforms $h$-orthonormal frame $\{e_i\}$ to $g$-orthonormal frame $\{a_i e_i\}$ for ...
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1answer
32 views

Curvature of a homogenous manifold.

I was a reading a paper and it seemed to me that in one of the equations the authors used the fact if $M$ is a homogenous Riemannian manifold (i.e., the group of isometries of $M$ act transitively on $...
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Flow of a vector field and harmonic oscillator

I'm trying to solve the following problem: Determine a vector field $X\in \mathfrak{X}(T\mathbb{R})$ whose flow gives the solutions of the equation of motion for the 1-dimensional harmonic oscillator:...
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Tubular-Neighborhood Theorems (proof question)

I have a question about part of the proof of the Tubular Neighborhood Theorem, which I believe reduces to a question about the $\epsilon$-Neighborhood Theorem. First some preliminaries for my ...
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72 views

Dual element of supporting function

(1) Assume that $C\subset \mathbb{R}^n$ is smooth, strictly convex, compact set and symmetric i.e. i.e., $x\in C$ implies $-x\in C$. Hence we have a norm $\| \ \|_C$ : $\| x\|_C=t >0$ where $\frac{...
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Hodge star operator on submanifolds

Let $M\subset (\Bbb R^3,g_0)$ be a surface (for example 2-sphere). Consider the following 1-form on $(\Bbb R^3,g_0)$: $$\omega:=ydx-xdy+zdz,$$ Then $\star\omega$ is a 2-form on $(\Bbb R^3,g_0)$. Now ...
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1answer
35 views

Glueing the metric

Let $(M,g)$ be a (smooth) compact Riemannian manifold with boundary. We glue the two copy of $M$, to make it closed Riemannian $M'$. Q 1.Can we make a modify $g$ near $\partial M$, such that there ...
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1answer
40 views

theorema Egregium and coeficcients of the second fundamental form

The theorema Egregium says that Gaussian curvature $K$ of a regular surface $S$ is invariant under local isometries. We have a local description of the Gaussian curvature as follows $$K = \dfrac{eg-f^...
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Show that $K=-\frac{\Delta \log \lambda }{\lambda ^{2}}$

Let $X:\Omega \subset %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$ conformal imersion, $K$ tha ...
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22 views

Connection coefficients on $S^3\subset(\mathbb R^4,\delta)$

I'm looking at Nakahara's Geometry, Topology and Physics, and the chapter on Riemannian geometry. In an example we consider $S^3$, which is parallelisable, so we can define a basis for $T_xS^3$ for ...
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Automorphism of unit hypersphere

I have been thinking what exactly the following map is: $$f(x)=\frac{x+\big\{\frac{(1-|p|^2)^{-\frac{1}{2}}-1}{|p|^2}<x,p>+(1-|p|^2)^{-\frac{1}{2}}\big\}p}{(1-|p|^2)^{-\frac{1}{2}}(<x,p>+1)...
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50 views

$\text{SL}(2,\mathbb{R})$ acts on the hyperbolic space by isometries

Let $H:=\{(x_0,x_1,x_2)\in\mathbb{R}^3\mid -x_0^2+x_1^2+x_2^2=-1\}$ be the hyperbolic space with metric $g_{hip}$ induced by the Lorenz inner product $g_{Lor}=-dx_0^2+dx_1^2+dx_2^2$. Find a bijection ...
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If a hypersurface has a self-intersection point, then the second fundamental form it is negative in a neighborhood of this point

I'm studying by myself Mean Curvature Flow and I'm reading Lecture Notes on Mean Curvature Flow by Xi-Ping Zhu. The doubt of the title of this topic arised when I read the proof of the following ...
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1answer
41 views

Exterior derivative calculation in Chern and Hamilton paper

Chern and Hamilton in their paper "On Riemannian metrics adapted to three-dimensional contact manifolds" constructed an structure on 3-sphere as follows (Example 3.2 of paper): Let $$\omega_1=...
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1answer
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Covariant Derivative of Covector Field

I'm trying to understand why the covariant derivative of a covector $\bf{\alpha} = \alpha_i \bf{e^i}$ is: $$\nabla_k\alpha = (\frac{\partial\alpha_i}{\partial x^k} - \alpha_j \Gamma^j_{ik})\bf{e^i} $$...
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1answer
55 views

Complete manifold in the Lee's book.

In the completeness charpter of book "Riemannian manifold: an introduction to curvature", he constructed two examples (at page 108) of non geodesically complete manifold. 1) any proper open subset of $...
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Why is euclidean geometry also called parabolic geometry?

Given that the three fundemental geometries are euclidean geometry (zero curvature), riemannian geometry (positive curvature) and lobachevskian geometry (negative curvature), I am curious as to what ...
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2answers
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Constant Gauss curvature $\Rightarrow$ homogeneous?

Let $S\subset \mathbb{R}^3$ be an embedded surface and $g_S$ the induced metric from $\mathbb{R}^3$ onto $S$. Since isometries preserve Gaussian curvature, $S$ homogeneous $\Rightarrow S$ has constant ...