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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Decomposition of forms in $\operatorname{SU}(4)$-manifold
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I ...
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Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
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Dirac operator on $\operatorname{Spin}(7)$, $G_2$ and $\operatorname{SU}(3)$ manifolds
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to ...
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Non-inertial frames of reference in empty space
Imagine that somebody wants to generalize special relativity to non-inertial frames of reference. For example I am going around a point and the metrics of space is non-Euclidean from my point of view. ...
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Making sense of constant scalar curvature metric horns
Suppose we have a compact oriented surface $S$ and we remove a point $p$ on it. We could consider a neighboorhood $U$ of the puncture $p$, so that the points in this neighboorhood are described by ...
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Upper bound on the sectional curvature of a Riemannian submersion
Consider the manifold $M := \operatorname{SO}(n) \times \mathbb{S}^{n-1}$, endowed with the product metric given by the bi-invariant metric of $\operatorname{SO}(n)$ and the round metric of $\mathbb{S}...
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Geodesic distance under conformal perturbation
Let $(M,g)$ be a complete Riemannian manifold of dimension $d\ge 3$. Suppose that $g_0$ is another Riemannian metric on $M$ which is conformal to $g$; i.e. $g = e^{2u}g_0$ for some $u\in C^{\infty}(M)...
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Manifold optimisation and likelihood [closed]
I have an estimation problem involving a likelihood function and missing/latent variables. These missing variables lie on a known geometric structure, specifically a manifold in the form of a complex ...
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Exhaustion function with uniform controls of level sets on universal covers of compact manifolds
recently I encountered the following problem in my research. Roughly speaking, it asks if, on the universal covers of a closed Riemannian manifold, one can find exhaustion functions with uniformly ...
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Laplace operators that give $S^d$ eigenvalues that are perfect squares
The Laplace-Beltrami operator on the sphere $S^d$ has eigenvalues $\{ k(k+d-1) : k \geq 0 \}$. Is there a geometrically natural Laplace operator / Laplace like operator (perhaps a Hodge Laplacian or a ...
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Isometry group of the Fubini-Study metric on complex projective spaces
Let $(\mathbb CP^n,g_{FS})$ be the complex projective space equipped with the standard Fubini-Study metric.
What is the Riemannian isometry group of $(\mathbb CP^n,g_{FS})$? It seems to me that its ...
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Hamiltonian approach to Einstein manifold theory
Let $(M,g)$ be a Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold. The zero section is denoted by $Z$.
We define a Hamiltonian on $T^0 M=TM\setminus Z$ via $$...
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Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
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Riemannian submersions and associated fibre bundles
My question is as follows, it is related to the chapter of Associated Fibre Bundles from [1].
Let $(X, g_X)$ and $(Y, g_Y)$ be two smooth manifolds and let $H$ be a Lie group which acts smoothly on ...
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Viscosity solutions of eikonal equation on Riemannian manifolds
It is well known that given a bounded open region $\Omega \subset \mathbb{R}^n$, the Dirichlet problem $$\lVert \nabla u \rVert = 1, \quad u|_{\partial \Omega} = 0$$
admits the unique viscosity ...
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