Questions tagged [riemannian-geometry]
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
3,083 questions
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Harmonic functions on complete Riemannian manifolds
I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at ...
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Decomposition of tensors
It is well known that every traceless, symmetric $2$-tensor can be decomposed uniquely into a Lie derivative part and a Codazzi part. Is there an analog for totally symmetric $k$-tensors?
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Solution existence in a pde system
If I have a smooth positive scalar function $f$ defined on a 2-dimensional manifold $M$, then $f:M\rightarrow (0, \infty)$, where the metric of $M$ is $g=\frac{dx^2+dy^2}{y^2}$, i.e., $M$ is Poincare' ...
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Decomposition about splitting of symmetric spaces of compact type
I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
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Questions about symmetric spaces
I'm a little confused with the following questions:
(1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$?
(2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...
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Question about Clifford volume element
I'm a little confused with the following:
Let $M$ be a $m$ dimensional Riemannian manifold and $e_1,\cdots,e_m$ be a local orthonormal base of $TM$. Let
$$
\omega_\mathbb{R}=c(e_1)\cdots c(e_m)
$$
...
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Why are we interested in spectral gaps for Laplacian operators
Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
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Is it known whether a closed simply-connected manifold of non-negative curvature admits positive Ricci?
It is discussed in this question whether a simply-connected closed Riemannian manifold with non-negative Ricci curvature admits positive Ricci curvature, and the answer appears to be "no, there ...
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Quasilinear second order parabolic equation
For the following parabolic equation
\begin{equation*}
\begin{split}
u_t&=\frac{u_{xx}}{1+u_x^2}-\frac{1}{u}\\
&u(x,0)=\cosh x+1.
\end{split}
\end{equation*}
How to show that $u(0,t)\sim \sqrt{...
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Homogeneous metrics on compact Lie groups
Given a differentiable manifold $M$, a Riemannian metric $g$ on $M$ is called homogeneous if its isometry group acts transitively on $M$. In that case, given any group $G$ acting transtitively on $(M,...
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Finding an asymptotically flat manifold with ${\rm Ric}_{r\phi} = \frac{\sin\theta}{r^2}$
Let $(r,\theta,\phi)$ be the spherical coordinates on $\mathbb{R}^3$ where $\theta \in (0,\pi)$ and $\phi\in (0,2\pi)$.
Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1$...
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Question about spin map
I'm confused with the following definition of a spin map.
A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...
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Midpoint geodesic polygon / Birkhoff curve shortening
I would like to know under what conditions the process
of creating a midpoint piecewise geodesic polygon converges
on a surface $S \subset \mathbb{R}^3$.
$S$ may be assumed smooth, closed, and ...
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Existence of a local spinor bundle
I am confused about the existence of a local spinor bundle.
My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...
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Manifolds and Riemannian geometry with a bundle viewpoint
I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ...
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Mapping torus of Klein bottle
This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO:
The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold.
The mapping class group of ...
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Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
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Non-negatively curved manifolds and the volume of balls
Whether a complete non-compact non-flat Riemannian $n$-manifold $M$ with non-negative sectional curvature has Euclidean volume growth?
That is, whether there is a constant $C>0$ such that $\mathrm{...
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Question about Dirac operator
Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that
$$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$
for $\...
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What are the volume-preserving diffeomorphisms of hyperbolic space? [duplicate]
What are the volume-preserving diffeomorphisms of $d$-dimensional hyperbolic space (in say the hyperboloid model)?
In particular, I'm especially interested in: what are the volume-preserving ...
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3 dimensional solvmanifolds and Thurston geometries
Does every three dimensional compact solvmanifold admit either Euclidean, nil, or sol geometry?
definitions/motivation/background:
A solvmanifold is a manifold $ M $ admitting a transitive action by a ...
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Relation between the distance projective maps and their angles
Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
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How negative Hessian of Hamiltonian implies strict convexity?
Let $X$ be a strictly convex domain in a Riemannian manifold $(\tilde{X}, g)$ of dimension $\geq 3$ with boundary defining function $\rho$ (so $\rho \in C^{\infty}(\tilde{X}), \rho>0$ in $X,<0$ ...
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conformal flat near a point
Let $(M,g)$ be a Riemannian manifold, and $x\in M$ be a fixed point.
Q Can we find a conformal transformation such that near $x$ we can write $e^{2u}g$ as $(dx^1)^2+\cdots+(dx^n)^2$?
Since the ...
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Positivity of an operator on a compact subset of a manifold
Let $E$ and $F$ be two vector bundles over manifold $X$. Let $P:\Gamma(E)\to \Gamma(F)$ be a self-adjoint differential operator over $X$. Define inner product on the spaces $\Gamma(E)$ of smooth ...
CommunityBot
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Sobolev norm of distance function on Riemannian manifold
$\DeclareMathOperator\SL{SL}$Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and ...
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On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
- 4,143
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Bubble tree convergence: Why is it necessary to consider centre of mass of the energy measure?
In the paper “Bubble Tree Convergence for Harmonic Maps” by Thomas H. Parker, after the picking the energy concentration points, he proceeded by expanding the map around each energy concentration ...
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Is there a geometric intepretation of the trace of tensor on a Riemannian manifold?
For a long time I thought the trace for matrices were just an elementary function with nice properties. But it is much more than that and really should be think of as a geometrical object (see ...
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Why doesn't this construction of the tangent space work for non-Riemannian metric manifolds?
In the 1957 paper, On the differentiability of isometries, Richard S. Palais gives a way to construct the tangent spaces of a Riemannian manifold using only its metric space structure (Theorem, p.1).
...
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Application of Santalo’s formula
Suppose that $(M,g)$ is a compact smooth Riemannian manifold with a smooth boundary and suppose that $f$ is a smooth function on $M$ with the property that
$$ \int_I f(\gamma(t))\,dt=0,$$
for any ...
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Bending surfaces in Riemannian manifolds
Let $S$ be an immersed surface in $\mathbb{R}^3$ (with the flat metric). We will call it flexible if there exists a smooth (or whatever regular) family of immersions $s_t: S\to \mathbb{R}^3$, such ...
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Nearly constant curvature implies "nearly isometric" to a space form?
It is well known a Riemannian manifold with constant sectional curvature is a quotient of the Euclidean space, hyperbolic space or sphere. In particular we know how their metric looks like locally.
...
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Reference request - Texts on geometric analysis with exercises
I’ve recently been studying some Riemannian geometry and geometric analysis, however I have found it difficult to find resources with exercises to practice. It seems that many textbooks past the ...
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Completeness on the tangent bundle
I was wondering if geodesics are defined for all time on compact Finsler manifolds, or more generally, for any spray on a compact manifold (where by geodesics, I simply mean the integral curves of the ...
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Low boundary of $\mathcal W$ function
Picture below is from Topping's Lectures on Ricci flow.
I don't understand the red line. From Lemma 8.1.8, I can get that $\mathcal W (g,f,\tau)$ has low boundary for any compatible $f,g,\tau$. But ...
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Divergence as infinitesimal volume change on a Finsler manifold
Let $M$ be a smooth manifold and $Z$ a smooth vector field on it.
It generates a family of diffeomorphisms $\phi_t:M\to M$ by demanding that $\phi_0=\operatorname{id}$ and $\partial_t\phi_t(x)=Z(\...
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What happens if in Randers metric the norm of the wind is not less than 1
One way to define the Randers metric is using the data $(h,W)$ associated to the Zermelo problem. Here $h$ is the Riemannian metric and $W$ is the wind. In order to define the Randers metric we must ...
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Exponential map for non-smooth Finsler manifolds
Context
I'm interested in studying reversible Finsler manifolds which do not have the strong convexity of the Hessian property (that is the Finsler function is a regular norm on every tangent space). ...
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Angle between two vectors in a Minkowski (Finsler) space
Given a Minkowski (or Finsler) space $(V,F)$, I am wondering how to define the angle between two vectors $w$ and $v$. I first thought it must be as $$\cos\theta(w,v)=\frac{g_w(w,v)}{\sqrt{g_w(w,w)g_w(...
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The differentiability of the distance function on asymptotically flat manifolds
Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball.
Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard ...
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Smoothness of some power of the geodesic distance in a Finsler geometry
I know that generally the geodesic distance $d_x$ from a point $x$ in a Finsler space is not smooth ($C^\infty$). According to Shen, the square of it is just $C^1$ at $x$. Now I am wondering if there ...
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Existence of geometric tubular neighborhoods in Finsler spaces
$\DeclareMathOperator\Tub{Tub}$I have not found any reference among the well-known books about the existence of a geometric tubular neighborhood in the Finsler spaces. I am wondering if there exists ...
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Volume form induced by a Finsler metric
I'm interested in knowing more about the volume form canonically induced by a Finsler metric.
I've found some reasoning about it in this article http://www.ams.org/journals/bull/1950-56-01/S0002-9904-...
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Smoothness of the asymptotic parametrization of a ruled surface
Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form
$$\begin{align}
\sigma \colon I \...
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Is the bundle map of the Eguchi-Hanson metric a Riemannian submersion?
Background. (Can be skipped if you already know what is the Eguchi-Hanson metric.) The Eguchi-Hanson metric $g$ is a complete Ricci-flat Riemannian metric on the cotangent bundle of the 2-sphere, $T^*...
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How is this product of tensors defined?
I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...
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Can we make distances in a finite subset of a manifold whatever we want?
Given a connected smooth manifold $M$ of dimension $m>1$, points $p_1,\dots,p_n\in M$ and positive values $\{d_{i,j};1\leq i<j\leq n\}$ satisfying the strict triangle inequalities $d_{i,j}<d_{...
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Equidistant points on a compact Riemannian manifold
Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows:
$K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at ...
- 10.6k
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votes
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Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
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