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.
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What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold?
For a connected, finite graph $G$, let $\lambda_1, \ldots, \lambda_n$ denote the nonzero eigenvalues of the graph Laplacian. We define $\zeta_G = \Sigma_{i = 1}^n \lambda_i^s$.
Then Kirkoffs Matrix-...
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Most natural connection on Lie group: comparison of different pictures
Let $G$ be a Lie group (not necessarily compact). One can equip $G$ with the left invariant metric (or
right invariant but in general there is no biinvariant metric in the noncompact case). Once the ...
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The first eigenfunction of Dirac operator for surface
Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar ...
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Morrey & Grauert - real analytic vector bundles admits analytic Riemannian metric
In theorem 1.2 of Brian Conrad's handout Operations with Pseudo-Riemannian metrics, the author writes
Theorem 1.2. Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \...
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Totally convex, convex and locally convex sets
Consider the following definitions :
$C\subset M$ is convex if any $p,\ q\in C$ all minimizing
geodesic between $p$ and $q$ are in $C$
$C$ is totally convex if for $p,\ q\in C$, every geodesic ...
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Clarification on Étienne Ghys' "Feuilletages riemanniens sur les variétés simplement connexes" paper
I apologize for this type of question, but I'm having some trouble to understand remark 3.4(4) on page 212 of this article, that reads
The restriction of $\overline{\mathcal{G}}$ (the foliation ...
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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 ...
user85365
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Action generated by geodesic flow is hamiltonian
I'm trying to understand why a certain action of a Lie Group is hamiltonian.
Let $(M,g)$ be a geodesically complete Riemannian manifold.
Then there exists a canonical one-form on the cotangentbundle $...
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The complex heat kernel on a Riemann manifold
There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial u}{\...
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Proof of the general expression for anomaly in a CFT and its partition function
I think the statement is that for any dimensional CFT the following is true,
$$\langle T^{\mu}_\mu \rangle = \sum B_n I_n - 2(-1)^{d/2}AE_d,$$
where $E_d$ is the `"Euler density" and $I_n$ are the ...
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Existence of Geodesics in continuous metrics
I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the ...
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Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
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Lower bound on volume of minimal hypersurface contained in a unit ball with curvature bounds
I was just wondering, if I have a geodesic ball of radius one in a manifold M whose sectional curvature lies between -epsilon and epsilon for epsilon small, and the injectivity radius of my manifold ...
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Manifolds with nonpositive radial curvature
How can one construct examples of Riemannian manifolds which have nonpositive radial curvature about some point, but are not nonpositively curved everywhere? (I presume that they exist, but do not ...
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Lower bound on the first eigenvalue of the Lichnerowicz Laplacian on positive Einstein manifolds
Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless ...
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Improving regularity of the boundary of a convex set in Riemannian manifolds
Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don't think it matters). Given a closed, convex subset $K \subset X$...
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The distance to the zero section of $TM$
Let $(M,g)$ be a Riemannian manifold. Let $S_g$ be the corresponding Sasaki metric on $TM$. For every $p\in M$, $V_p\in T_pM$, is it true and obvious that $0_p$ is the closest point of the zero ...
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Cohn-Vossen rigidity theorem in hyperbolic space
There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces in ...
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Proof that geodesics have zero curvature with Cartan's moving frames method
I tried to use Cartan's moving frames method to prove that any minimizing-length curve in $\mathbb{R}^2$ has zero curvature. Here below is my idea of proof. I am asking for a proof-verification and ...
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Quantitative upper bound on mean curvature of an isometric embedding
By Nash embedding theorem, any complete Riemannian manifold $M$ can be isometrically embedded in $\mathbb{R}^N$, for sufficiently large $N$.
The proof of the theorem is quite involved, and it is not ...
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The Hausdorff dimension of the union of singular orbits and exceptional orbits
Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
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Functional decaying under the heat flow (?)
Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$.
For any positive function $v$, I set
$$
J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g.
$$
...
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Second fundamental form blows up at minimal hypersurface singularity
I have seen the claim that "$A$ bounded on an area minimizing current implies no singular set" in a couple of papers by Lohkamp, but with no reference (see https://arxiv.org/abs/1805.02180 e.g.).
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Question on Weil-Petersson metric on Teichmuller space
I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space".
The tangent space at ...
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sequence of graphs converge in the sense of varifold to multiplicity 2 plane
Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
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Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold
I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem:
Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...
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Gaussian Curvature of Exponentiated 2-Planes
Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...
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The "Rolle theorem" for sections of a vector bundle
1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
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rigidity of eigenvalues of circular ensemble
Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...
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Sobolev embedding on sphere
Let $S$ be a two-dimensional sphere, $\Delta$ be the Laplace-Beltrami operator on $S$ and $L^p(S)$, $p\geq 1$, be the usual $L^p$ space of real-valued functions on $S$. We also set $\|f\|_{H^\alpha(S)}...
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Proof of equivalence between Lie triple systems and totally geodesic submanifolds
In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a ...
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3-manifolds with all minimal surfaces closed
Question. Let the manifold $(M^3,g)$ be compact without boundary. Suppose that every complete, embedded minimal surface $\Sigma \subset M^3$ is closed. Must $M$ be diffeomorphic to $\mathbf{S}^3$ or $\...
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Geometric invariants of a Riemannian manifold encoded in certain moment map
Let $(M,g)$ be a Riemannian manifold with isometric group $G=Iso(M,G)$. The metric gives an isomorphism between tangent and cotangent bundle of $M$. So $g$ induce a natural symplectic structure on $...
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"Isoperimetric inequality" for self intersecting closed surfaces?
As the title suggests, I am trying to find something like an isoperimetric inequality for smooth immersions of the 2-sphere into $\mathbb{R}^3$ that relates the surface area to the enclosed 3d-volume. ...
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An alternative representation of the principal symbol of the Laplace operator
Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure.
Are the following two conditions equivalent?
First condition ...
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Gaussian bounds for the heat kernel of regular domains in Riemannian manifolds
In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same ...
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Connection between Gram matrix and Riemannian invariants?
Recall that the Gram matrix of vectors $v_1, \dots, v_k\in\mathbb{R}^n$ is the $k\times k$ matrix $G_{ij}=(v_i,v_j)$. Now suppose that the vectors $v_i$ have been sampled uniformly from some ...
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The radially symmetric isoperimetric problem
A solution to the $l$-isoperimetric problem on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant:
$$h(\gamma) = \frac{l}{...
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Convergence of Riemannian metrics spectra
Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
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Generalized Hawking Mass
This is a fairly general question. Let $(M^3,g)$ be a Riemannian 3-manifold. Let $\Sigma^2$ be a dimension-2 submanifold of $M$. The Hawking mass of $\Sigma^2$ is defined as
$m(\Sigma^2) := \frac{|\...
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Symmetries vs. Bound in codimension of Nash isometric embedding
Let $(M^m,g)$ be a compact smooth Riemannian manifold of dimension $m$. From the celebrated Nash Embedding Theorem, we know there exists a (smooth) isometric embedding $M\hookrightarrow\mathbb R^n$ on ...
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Convexity and Strong convexity of subsets of Surfaces
In the book Riemannian geometry - modern introduction by Isaac Chavel, three different definitions of convexity are introduced. I am looking for an example of a set which is convex but not strongly ...
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Tetrad postulate: Implies or results from the metricity of the connection?
Hi,
I see that the tetrad postulate:
$\nabla_{\mu}e_{\nu}^{I}=\partial_{\mu}e_{\nu}^{I}-\Gamma_{\mu\nu}^{\rho}e_{\rho}^{I}+\omega_{\mu J}^{I}e_{\nu}^{J}=0$
Can be merely derived from writing a ...
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Monopole classes on hyperbolic 3-manifolds
Let $M$ be a closed hyperbolic $3$-manifold, and $e \in H^2(M)$ an integral cohomology class which is the first Chern class of a $Spin^c$ structure on $M$. Suppose there is a solution to the monopole ...
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A (possible) Lie algebra extension of the Lie algebra of a foliation
Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is ...
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Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
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Is every linear Lie group of bounded geometry?
$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...
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What normal variational vector fields are allowed for the area to be preserved?
Let $(M^3,g)$ be a Riemannian manifold with boundary and let $\varphi : \Sigma \to M$ be a two-sided proper embedding of a compact surface with boundary, with unit normal $N$.
If $\varphi$ is not ...
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Möbius strip zero curvature [closed]
Is there a Möbius strip, seen as an embedded surface in $\mathbb{R}^3$, with zero curvature? I know one can see the Möbius strip as the quotient of the square with reverse identification of two sides ...
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Intersections of geodesics in an "almost flat" plane
Let $g$ be a complete metric on $\mathbb{R}^2$, such that:
Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes.
The integral of the Gaussian curvature in $K$ is ...
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