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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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Is there a pattern for closed and co-closed $n$-forms on $\mathbb{R}^{2n}$?

Consider $\mathbb{R}^{2n}$ with its standard Euclidean Riemannian metric. Let $\omega \in \Omega^n(\mathbb{R}^{2n})$ be an $n$-form. I am trying to understand if there is a succinct way to express ...
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Computing Fubini-Study metric from the formal definition

Definition: the Fubini-Study metric $g_{FB}$ on $\mathbb{CP}^n$ is the only metric which makes the projection $\pi:(\mathbb{S}^{2n+1},g)\to(\mathbb{CP}^n,g_{FB})$ a Riemannian submersion (where $g$ is ...
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Why does the exponential map have this form?

I believe I understand what the exponential map does, in informal terms. However I cannot relate this to the equation I see before me in the papers. The exponential mapping function for a symmetric ...
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the smoothness of an induced orthonomal vector field in normal neighborhood

Let $M$ be a Riemannian manifold and p be a point on M. Let U be a normal neighborhood about p. Fix a vector $v_p∈T_pM$.And use parallel transport,we can transport is to $T_qM$ for any $q\in U$,...
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Understanding iterated covariant derivatives to define Sobolev spaces on manifolds

I'm having big troubles understanding the definition of Sobolev spaces on manifolds. Ok, so we have a Riemannian manifold $(M, g)$, and then we can define a natural riemannian measure (which I will ...
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25 views

Smoothness of a vector field defined by a system of differential equations

There is a lemma stating that there exists a unique vector field $G$ on the tangent bundle $TM$ of a manifold $M$ whose integral curves are of the form $t\mapsto (\gamma(t),\gamma’(t))$ where $\gamma$ ...
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28 views

Christoffel symbols of the second kind transformation law

We want to show that the Christoffel symbols of the second kind transform like a connection. the Christoffel symbols of the second kind are given by: $$\begin{Bmatrix}a \\ bc\end{Bmatrix} = \frac{1}{...
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14 views

Local Automorphisms of Cartan Geometries are determined by values at a point

Let $M$ be a manifold and $(P\to E\overset\pi\to M, \omega: TE\to\mathfrak g)$ a Cartan geometry on $M$. I have seen the following statement: Let $f:U\to V$ be a local automorphism where $U,V$ are ...
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Comparison of volume forms on Riemannian manifolds.

I am reading the Cheng's paper (1975), which states that Theorem. Suppose $M$ is a complete Riemannian manifold and Ricci curvature of $M\geq(n-1)k,\ n=\mathrm{dim}\ M.$ Then, for $x_0\in M$ we have $...
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Holomorphic forms: a very basic notion

Let $M$ be a holomorphic (complex) manifold with: $$(1)\quad dim_{C}(M)=m. $$ What I understand regarding a holomorphic form is that it is: $$(2)\quad \alpha^{(r,0)}=\frac{1}{r!}f_{\mu_1,\dots,\mu_r}...
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Non-trivial explicit example of a flat connection

We all know that the exterior derivative on the trivial bundle forms an example of a flat connection. Can anyone provide an explicit example of a flat connection that is not just the exterior ...
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Zero constant mean curvature in Minkowski space versus in Euclidean space

There's a famous result in $\mathbb{R}^3$ which goes as: there are no compact minimal surfaces in $\mathbb{R}^3$.So the mean curvature cannot be zero in compact surfaces in $\mathbb{R}^3$. Now what's ...
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Is it true that any Riemannian metric on cylinder can be deformed to product metric?

Is it true that any Riemannian metric on cylinder $\Bbb S^1\times \Bbb R$ can be deformed to standard product metric? Are there some standard references about such classifications?
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Use of the Rellich Lemma in the proof of the Hodge Theorem

I am reading through the proof of the Hodge theorem that is given in Griffiths' and Harris' Principles of Algebraic Geometry, see pages 84-100. The method of proof is to establish a weak solution of ...
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Geodesics and a general pregeodesic equation

Let $(M,g)$ be a Riemannian manifold, and let $\nabla$ denote the Levi-Civita connection. Then we say a smooth curve $\gamma:J\to M, t\mapsto\gamma(t)$ is a geodesic if $$D_t\gamma'=0.$$ We say a ...
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35 views

Does ${\rm Iso}(M\times N)={\rm Iso}(M)\times {\rm Iso}(N)$ hold for product metrics?

I know that ${\rm Iso} (M\times N)={\rm Iso}(M)\times {\rm Iso}(N)$ is not generally correct where $M$ and $N$ are smooth Riemannian manifolds; but Does ${\rm Iso} (M\times N)={\rm Iso}(M)\times {\...
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20 views

If $(e_1, \ldots, e_n)$ and $(f_1, \ldots, f_n)$ are co-oriented, then so are $(e_1, \ldots, e_n, u)$ and $(f_1, \ldots, f_n, u)$

Suppose that $S$ is a smooth $n$-submanifold of $M$ where $\dim(M) = n+1$. Suppose also that $S$ and $M$ are both Riemannian and oriented. Suppose that $(e_1, \ldots, e_n)$ and $(f_1, \ldots, f_n)$ ...
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50 views

Hodge-$\star$ operator computation on a smooth two-dimensional manifold

Let $(x,y)$ be the local coordinates on a Riemannian manifold $M$ with $\dim(M) =2$. Let $\star$ denote the Hodge-$\star$ operator, and let $g = g_{ij}$ denote the Riemannian metric on $M$. I am ...
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48 views

On the notion of tensor in Riemannian Geometry

In DoCarmo’s Riemannian Geometry, a tensor of order $r$ on a Riemannian manifold is defined as a multilinear mapping $$T:\Xi^r(M)\rightarrow C^{\infty}(M)$$ where $M$ is a smooth manifold of dimension ...
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23 views

Well-definedness of assinging a value to a point of a manifold with respect to parallelism

By definition as in DoCarmo’s Riemannian Geometry book, in a smooth manifold $M$ a vector field $V$ along a smooth curve $c:(a,b)\rightarrow M$ is a mapping $t\mapsto V(t)\in T_{c(t)}M$ such that $t\...
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36 views

Metric on $M/G$ which makes $\pi: M\to M/G$ a Riemannian submersion

Let $G$ be a Lie group acting isometrically, freely and properly on a Riemannian manifold $(M,g)$ and $\pi:M\to N:=M/G$ the natural projection. Show that there is a unique metric $h$ on $N$ which ...
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What exactly this “elliptic” and “hyperbolic” mean on the picture describing Lobachevskian and Riemannian geometry?

Item 5 here has a figure calling Lobachevskian geometry hiperbolic and Riemann geometry elliptic and both figures have a perpendicular line. What does that perpendicular line and the both pictures ...
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Why is the metric on a hyperboloid different than the metric on a sphere?

I am talking about the unit sphere and unit hyperboloid in $\mathbb{R}^3$. To get a metric you might use the riemann metric and the length of a curve. To calculate the length of a curve $\gamma$ from $...
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24 views

Extending to a local frame that agrees with given orientation

Suppose that $(e_1, \ldots, e_k)$ is an oriented basis for $T_pM$ where $M$ is an oriented Riemannian manifold. In general, we know that we can extend to a smooth local frame $(X_1, \ldots, X_k)$ on $...
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An upper bound of the first eigenvalue of Laplacian on a Riemannian manifold.

I'm reading the Cheng's thesis ""Eigenvalue Comparison Theorems and Its Geometric Applications," and the author obtains an estimate of eigenvalues of the Laplacian based upon his theorem: If $M$ is $...
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18 views

(Total) volume preserving transformations?

We're looking at some compact Riemannian (maybe pseudo-Riemannian) manifold ($M$) without boundary and metric $g$. How would I go about finding the possible Weyl rescalings $w(x^{\mu}$) of the metric: ...
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1answer
41 views

Levi-Civita connection for a metric in $\mathbb{R}^{3}$

Let $g$ be a metric in $\mathbb{R}^{3}$ defined as $\partial_{x}, \partial_{y}, \partial_{z}$ are orthogonal everywhere, and $g(\partial_{y},\partial_{y})=1, g(\partial_{z},\partial_{z})=f(x), g(\...
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25 views

How/why does the contraction of standard volume form give the canonical form.

$M \subset \mathbb{R}^{N}$ is a (oriented) $n-1$ dimensional submanifold. Suppose $\nu \in T_{p}M^{\bot}$, of length one (a normal unit vector on $M$). How and why does the contraction $\nu_{\neg}(...
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Finding Christoffel Symbols using via variational method.

I'm trying to find the Christoffel Symbols for the Lorentz metric $${\rm d}s^2 = \cos(2\pi x)({\rm d}x^2-{\rm d}y^2) - 2\sin(2\pi x)\,{\rm d}x\,{\rm d}y$$by looking at the Euler-Lagrange equations for ...
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37 views

Inconsistent statements about Laplacian versus Laplace-Beltrami

The Laplace-Beltrami operator is said to be intrisic: it can be defined in terms of the metric and without reference to the "ambient" coordinate system. Not so for the Laplacian, it is defined in ...
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Ricci flat symmetric spaces are flat?

It is stated in Joyce's 'Compact Manifolds with Special Holonomy' (P.124) that if $M$ is a compact Riemannian symmetric space, then $M$ is Ricci flat implies $M$ is flat. I am having trouble seeing ...
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38 views

Is there any non-complete metric on $\Bbb R^2$?

I've read several books with this content: Let $g = dr^2 + f^2(r)d\theta^2$ be a smooth metric on $\Bbb R^2$ expressed in polar coordinates. This metric is complete and the volume can be finite ...
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Is there such a thing as a “most parallel” vector on a curved manifold?

My question is in response to Dirac's assertion: Suppose we have a vector $\left\{ A^{\mu}\right\}$ at a point $\mathscr{P}$. If the space is curved, we cannot give meaning to a parallel vector at ...
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Is the “immersed proper” hypothesis necessary in Half-space Theorem?

I'm using the following version of Half-space Theorem: $\textbf{Theorem}$(Half-space) A connected, immersed proper, nonplanar minimal surface $M$ in $\mathbb{R}^3$is not contained in a ...
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Evolution equation of radial function of a star-shaped hypersurface

Let $F_t:\Sigma^n\to\mathbb{R}^{n+1}$ be a smooth family of immersion of a hypersurface that satisfies the inverse curvature flow: \begin{align} \tag1 \frac{\partial F_t}{\partial t}=\frac{1}{f}\nu. \...
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Cogeodesic flow on compact manifold has compact leaves and manifold

Let $g_{ij}$ be positive definite metric on a compact manifold $M$. Consider hamiltonian $H=\sum_{ij}\frac{1}{2}g_{ij}(x) p_ip_j$ with $p_i\in\Gamma(T^\star M)$. Then define $E_\lambda=\{(x,p)\in T^\...
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Help to understand that $x\left( z\right) ={Re}\int_{z_{0}}^{z}\phi \left( \xi \right) d\xi $ is well defined

Let $\phi :\Omega \subset %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion \rightarrow %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $ holomorphic function. Define $x:\...
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Symplectic connections are (locally) Levi-Civita connections

I was wondering... Is every symplectic connection $\nabla$ on some symplectic manifold $(M, \omega)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?
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Einstein: Is invariance of the sum of squares of coordinate differences of a rigid separation sufficient to determine Euclidean space?

I didn't include my source in my original version of this posting because I didn't want to get an answer that was out of deference to the the source rather than based on reason alone. But I didn't ...
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Hyperbolic Geometry: Representation of the metric in a chart

Let $\mathbb{R}^{1,n}$ denote Lorentzian $n$-space, i.e., $\mathbb{R}^n$ equipped with the Lorentzian inner product $$\langle x,y \rangle = - x_0 y_0 + x_1 y_1 + \cdots + x_n y_n.$$ Define $\mathbb{H}...
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Maximal smooth atlas of an open subset $U$ of a smooth manifold $M$

Let $M$ be a smooth $n$-dimensional manifold, with smooth maximal atlas $\mathcal{A}:=\{\varphi:O_a \rightarrow \varphi(O_a)| a \in A\}$. If I take an open subset $U \subset M$, and endow it with the ...
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If two Riemannian manifolds are isometric as metric spaces, must they be isometric as Riemannian manifolds?

In other words, is it possible that distinct Riemannian metrics can induce the same metric space structure on a smooth manifold?
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1answer
30 views

Concrete interpretation of parallelism of a vector field along a curve

Let $M$ be a smooth manifold with an affine connection $\nabla$. A vector field along a curve $c$ is called parallel if its covariant derivative $\frac{DV}{dt}$ along $c$ is equal to $0$. This notion ...
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27 views

Is velocity field extendible to a vector field on the whole manifold?

In DoCarmo’s Riemannian Geometry book, an affine connection $\nabla$ on a smooth manifold $M$ is defined to be a mapping :$\Xi(M)\times \Xi(M)\rightarrow \Xi(M)$, where $\Xi(M)$ is the set of all ...
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33 views

Show that any affine connection $\nabla$ on $\mathbb{R}^n$ is of the form $\nabla=D+\Gamma$.

Show that any affine connection $\nabla$ on $\mathbb{R}^n$ is of the form $\nabla=D+\Gamma$, where $D$ is the Euclidean connection and $\Gamma:\mathcal{X}(\mathbb{R}^n) \times \mathcal{X}(\mathbb{R}^n)...
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1answer
23 views

A relation between left-invariant vector fields

If $G$ is a Lie group with a bivariant metric and if $U,V,X$ are left invariant vector fields, I wish to prove that $\langle[U,X],V\rangle=-\langle U,[V,X]\rangle$. Following the proof of Do Carmo’s ...
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1answer
19 views

Exponential map on the Fisher manifold for exponential family distribution

So, I don't really understand too well Diff. Geometry and Manifolds currently. Hence, I've started studying it and it is very interesting. However, atm I just need to understand how to compute the ...
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39 views

Homogeneous polynomial on unit sphere.

Suppose $ F $ is a homogeneous polynomial function of degree $ m $ on Euclidean space $ \mathbb{R}^{n+1} $. Restrict $ F $ to the unit sphere $ S^n $ and we get a function on $ S^n $ denoted by $ f $, ...
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Meaning of commutativity of a local flow of a vector field and left-translations

In DoCarmo’s Riemannian Geometry book, it is writen that if $x_t$ is the flow of a left invariant vector field $X$ of a lie group $G$, then $L_y\circ x_t=x_t \circ L_y$. But the domains of the left ...