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Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

5,934 questions
34 views

A non-vanishing vector field on $S^3$ whose flow does not preserve any transversal foliation

Is there a non-vanishing vector field $X$ on $S^3$ which does not admit a transversal $2$-dimensional foliation? if the answer is negative, is there a non-vanishing vector field $X$ on $S^3$ which ...
60 views

Irrational closed orbits of vector fields on $S^2$ (Limit cycles and trace formula)

Motivations: We first introduce our motivations: We wish to find an operator-theoretical interpretation for the number of limit cycles of a polynomial vector field on the plane. Via Poincaré ...
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Projection of an invariant almost complex structure to a non-integrable one

My apologies in advance if my question is obvious or elementary. We identify elements of $S^3$ with their quaternion representation $x_1 + x_2i + x_3j + x_4k$. We consider two independent vector ...
41 views

Radius of curvature question [on hold]

Im trying to figure out an equation from geometric geodesy. But doing the derivation of that equation results in a different equation then the one from the literature. Its the (1-e^2). I constantly ...
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Existence of nonparabolic ends

Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the ...
96 views

Asymptotic bound on minimum epsilon cover of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...
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Hermitian sectional curvature

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair. ...
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Subdividing a Compact Bounded Curvature Manifold into Charts with Bounded Lipschitz Constant

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$ cover $P$ of $M$; that is ...
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Is there a representation theoretic way to define the pullback of densities and differential forms?

I find it convenient to define the bundle of densities of weight $\alpha$,say $\Omega_\alpha(M)$ over a smooth manifold $M$ as the associated vector bundle of the frame bundle $F(M)$ with the ...
123 views

On the definition of the Reeb foliation

To define the Reeb foliation on the sphere, one needs to fix two even functions of the from $f:(-1,1)\to\mathbb{R}$. In the book I. Tamura, Topology of Foliations: An Introduction, the following is ...
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An algebraic foliation chart for a foliated manifold is a foliation chart for which the transition maps are polynomial maps. What is an example of an analytic foliation of the Euclidean space $\... 0answers 121 views Why not hermitian metrics on the tangent bundle? Let$(M,J)$be an almost complex manifold. Then$TM$is naturally a complex vector bundle. Hence it makes sense to consider a hermitian metric$h: TM \otimes \overline{TM} \rightarrow \mathbb{C}\$, ...

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