Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
9,385 questions
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Is "asymptotic dilation equivariance" a known hypothesis for one-parameter families of maps $\mathbb R^3 \to \mathbb R^3$?
Let $X : \mathbb{R}^3 \times (0,\infty) \to \mathbb{R}^3$, write $X_\lambda = X(\cdot,\lambda)$, and suppose each $X_\lambda$ is $C^2$ and smooth in $\lambda$. Let $S_\lambda$ denote the set of corank-...
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Reference request: a pointwise boundary identity for one-parameter families of free boundary and capillary minimal hypersurfaces
Let $(M^{n+1},g)$ be a Riemannian manifold admitting a distance function $\rho$ with $|\nabla\rho|=1$ whose level sets $S_r$ are umbilic, i.e. $\mathcal{A}=\lambda\, g|_{TS_r}$ for the second ...
- reputation score 79
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Geometric significance of this space of invariant 2-forms on $S^5$
$\DeclareMathOperator\SU{SU}$Consider $S^5=\SU(3)/\SU(2)$ with the $\SU(3)$ action, and the space of invariant 2-forms under this action. If I'm not mistaken the usual invariant Cartan 3-form $\omega$ ...
- reputation score 37539
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Spectral gap and shortest dual vectors for flat tori generated over the multiquadratic field $K = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$
I am studying the spectral properties of the Laplace-Beltrami operator on a specific class of compact Riemannian manifolds constructed via arithmetic lattices.Consider the manifold $M = S^4 \times T^3$...
- reputation score 47
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Minkowski's existence theorem for polytopes with mixed causal facet normals?
Minkowski's classical theorem states that, given unit vectors n₁, ..., nₖ ∈ ℝⁿ (Euclidean) that span ℝⁿ and positive weights w₁, ..., wₖ satisfying the closure relation ∑ᵢ wᵢ nᵢ = 0, there exists a ...
- reputation score 41
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Quantization of harmonic periods on product manifolds $T^2 \times T^2$ under vanishing off-diagonal form decomposition
Summarize the problem
I am investigating the period integrals of harmonic forms on a smooth 4-manifold with product topology $M \cong T^2 \times T^2$, equipped with real angular coordinates $(\theta_1,...
- reputation score 7
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156 views
A finer Birkhoff ergodic theorem in terms of rate or order of convergence of the average measure
Let $f$ be a measure preserving and ergodic dynamical system on a finite measure space $(M, \lambda)$.
According to Birkhoff ergodic theorem, for a given measurable function $\phi:M\to ...
- reputation score 642
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64 views
Calculating degree of unit tangent field on a 1-dimensional submanifold
I am currently reading a paper from R. Lutz Structures de contact sur les fibrés principaux en cercles de dimension trois which can be found on the link here:
https://www.numdam.org/item/?id=...
- reputation score 41
Score of 12
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454 views
Do we have any "concrete" examples of Nabutovsky-Weinberger metrics?
For brevity, say that an NW-metric is a Riemannian metric $g$ on a smooth compact manifold $M$ with $dim(M)\ge 5$ such that $g$ is a local minimum for the diameter functional $Diam$ on the space of ...
- reputation score 20225
Score of 3
1 answer
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Distribution integrable in one point implies integrable in neighborhood?
Let $M$ be a smooth manifold. Let $D\leq TM$ be a subbundle i.e. a distribution of rank $k$. Then $D$ is integrable at $p \in M$ if there is a submanifold $N \subseteq M$ with $p \in N$ and $T_xN = ...
- reputation score 1010
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Existence of Morse function on flat manifolds for which the Hessian of singularities of the same index are parallel related
Let $M$ be a flat simply connected manifold.
Does $M$ necessarily admit a Morse function $f:M\to \mathbb{R}$ such that every two singular points $p,q$ with the same index have ...
- reputation score 642
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198 views
Cohen, Jones and Segal's definition of flow category
I am reading the paper "Morse theory and classifying spaces" by Cohen, Jones and Segal. I am well aware (see for instance this post on the blog by Brazelton, Calle, Gould and Mejia) that ...
- reputation score 163
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What's the point of flow categories
In the paper "Morse theory and classifying spaces", Cohen, Jones and Segal define the flow category $\mathcal{C}_f$ of a Morse-Smale function $f: M \to \mathbb{R}$. It is a topologically ...
- reputation score 163
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78 views
Do complete non-rigid gradient Schouten solitons exist?
Let $(M^n,g)$, $n\geq 4$, be a complete Riemannian manifold. A gradient
Schouten soliton is a pair $(g,f)$ satisfying
$$
\operatorname{Ric}+\nabla^2 f
= \left(\frac{R}{2(n-1)}+\lambda\right)g,
$$
...
- reputation score 981
Score of 6
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114 views
Has anyone put a scalar field theory on a diffeological space?
I’m trying to figure out if anyone has actually built a scalar field theory (action, equations of motion, maybe quantization) directly in the diffeology framework, rather than just using diffeology to ...
- reputation score 69
