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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Invariant Lagrangians of a connection and its derivatives: how do they look like?
Let
$$
L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma)
$$
be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...
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Projection of geodesic is geodesic
Background : If a compact Riemannian manifold $M$ with a no curvature condition has disjoint two submanifolds $N_i$, then the distance between them is attained by some minimizing geodesic $c$.
If $c'(...
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Some manifold which is not totally geodesic in a compact manifold
(1) If $N^k$ is a submanifold in a compact Riemannian manifold $M^{k+m},\ m\geq 1$ s.t. each $p\in N$ has the following property : There exists independent set $\{ X_i\}_{i=1}^k$ tangent to $T_pN$ s.t....
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Local differential geometry and invariant theory
Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists?
Start with ...
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Possible directions of saddle connections
Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal ...
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Dimension of tangent space to manifold of cross section slices
Given a function $\Phi:\Omega^{\Phi}\subset \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\Omega^{\phi}\subset \mathbb{R}^2\rightarrow\mathbb{R}$, using a ...
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Scattering in (pseudo-)Riemannian spaces
I will ask my question in a broad way, leaving a lot of freedom for answers.
Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
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Existence of left-invariant metric on the cotangentbundle of homogeneous spaces?
Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
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Closed geodesics avoiding points in hyperbolic surfaces
Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...
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Can a conformal map be turned into an isometry? [closed]
Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with
$$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, \...
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No normal coordinates on general Finsler manifolds
I recently read a footnote in Chern's article stating that a non-Riemmanian Finsler manifold does not possess normal coordinates.
As I'm still new to non-Riemmanian Finsler geometry I don't see why ...
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The injectivity radius of $L^2$ metrics
Suppose M is a compact manifold and Rim(M) the space of all Riemannian metrics.
Consider $L^2$ metric $ G_g(h,k) = \int_M tr(g^{-1}hg^{-1}k)vol(g)$ where is a Riemannian metric and $ h,k \in T_gM$ . ...
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Zero set of eigenfunction along a sub manifold
Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...
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Length spectrum and Zoll surfaces of revolution
The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed
geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces,
all of whose ...
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Are isospectral manifolds necessarily homeomorphic?
It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric.
Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic?...
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Some Confusion on Harmonic Map
I am stuck at Corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.
Corollary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}...
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Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds
For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details.
Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen (http://...
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Examples of two-dimensional Riemannian manifolds that can't be isometrically embedded into $\mathbb{R}^4$
Can anyone give some examples of two-dimensional Riemannian manifolds $(M,g)$ that can't be isometrically embedded into $\mathbb{R}^4$? (Further more Globally)
What if it is smooth?
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Does the tensor bundle of a compact manifold have a bounded geometry?
Let $M$ be a compact manifold. Let $S^2 T^*M $ be the vector bundle of all symmetric $(0,2)$ tensors and $S_+^2 T^*M$ be the open subset of all positive definite ones. Does $S_+^2 T^*M $ have bounded ...
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Construction of fibration over Riemannian Manifold
Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$.
More precisely:
I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some $C^{\infty}...
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The orthodrome of n-spheres.
I am a Computer Science undergraduate who does a lot of other tinkering in his free time. Right now, I'm tinkering with n-spheres. Specifically, I'm looking at the distances between a collection of ...
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Intuitive understanding of the mean curvature flow [closed]
I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = (...
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Hodge de Rham operator and orientability
Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
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Some questions on the nodal geometry of Dirac operators
Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:
Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
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Darboux-like coordinates on a Kähler manifold
If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...
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Christoffel symbols of a moduli of smooth curves
The Setting:
Let $H$ be the Hilbert space of all class $C^k$-curves into $\mathbb{R}$ with inner product: \begin{equation}
<f,g>:=\int_{\mathbb{R}} f'(x)g'(x) e^{-x}dx
\end{equation}
...
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Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?
$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
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Applications of Gauss-Bonnet theorem
In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra
using Gauss Bonnet theorem.
I think given how central it is to mathematics with its far reaching generalizations ...
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Carre du Champ, Subunit Paths and CC-metrics
Let the operator $L$ be given by $Lf(x):=\nabla\cdot (A\nabla f(x))$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ belongs to a suitable class of functions $\mathcal{A}$. The carre du champ operator $\...
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Curvature Characterization of Homogeneous Spaces
A (Riemnnian) symmetric space is locally characterized by the first covariant derivative of its curvature tensor, namely, a manifold $M$ is locally a symmetric space if and only if $\nabla R=0$ (where ...
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Parallel Transport on Hypersurface Spinor Bundle
So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link:
https://projecteuclid.org/...
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Heat kernel asymptotics for small distances
I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies
$$p_t(x, y) = \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\...
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Hodge map and the Cohomology Ring of a Riemannian Manifold
For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology ...
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stochastic control / geometric mean
Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
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Totally geodesic subgroups in Lie groups
Let $G$ be a Lie group with a left invariant metric $g$.
Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $...
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Obtaining Hessian of the embedding from an induced metric
Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...
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Regularity of the distance from the boundary in singular riemannian manifolds
I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds.
I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...
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Compact Riemann manifolds with constant injectivity radius
I'm interested in compact Riemann manifolds which have that property that the injectivity radius at a point $p$ doesn't depend on $p$. Another way to put this is that the function $$p \mapsto d(p, \...
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Flows associated with Killing fields
Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
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Hausdorff convergence of submanifolds in Riemannian manifolds
Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact ...
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Upper bound for Willmore energy
Good day to everyone! Does anybody know if there are upper bound estimates for Willmore energy for a given surface?
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Understanding the Exp map from a moduli of smooth curves
The setup:
Suppose I'm given a family of class $C^k$ curves $\{f_{z}\}_{z\in Z}$ with values in $\mathbb{R}$ parameter by a finite number of parameters $Z\subseteq \mathbb{R}^m$.
Let $\mathscr{M}$ be ...
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material derivative and relation to Riemannian metric
For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$.
Smooth functions on $N$...
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Uniform partitions of a compact Riemannian manifold
My question is a little bit vague. I want to know if an arbitrary compact Riemannian manifold (M^d,g) admits partitions that are uniform in some sense. To be more precise, I need for every eps > 0 a ...
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Differentiability of a map to the free loop space
While reading Morse theory, closed geodesics, and the homology
of free loop spaces, the author claims the following:
Given the $S^{n-1} \hookrightarrow Y_1 \rightarrow T^1S^n$ bundle over $T^1S^n$, ...
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The scope of correspondence principle in quantum chaos
My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...
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Stability for open manifolds of finite volume under lower Ricci curvature bound
By a theorem of Cheeger-Colding if $N_i\to M$, where $\to$ means Gromov-Haussdorff convergence, $N_i$ and $M$ are of the same dimension, compact and smooth Riemannian manifold with finite volume, as $...
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Constant Harmonic Mean surfaces
For surfaces embedded in $\mathbb R^3$ with principal curvatures $ \kappa_1, \kappa_2 $ we know bending/isometric mappings conserve $ K= \kappa_1 \kappa_2 $ and CMC DeLaunay type minimal surfaces ...
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Conformally flat with zero scalar curvature 2 [duplicate]
First, i am the one who asked about the existence of compact manifolds of dimension $n\geq 4$ which are conformally flat, non-flat, with zero scalar. Due to the fact that i couldn't comment because i ...
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A Converse to Cartan–Hadamard theorem?
Let $M$ be a complete Riemannian manifold, with the property that $\exp_p\colon T_pM \to M$ is a diffeomorphism for every $p \in M$.
Can we say something about it's curvature?
Is it true that its ...
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