Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,630
questions
7
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A cohomology associated to a vector field on a Riemannian manifold
Edit: Accoring to the comment of Asura Path I revise the question.
Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...
12
votes
1
answer
663
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Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach
Let $P\rightarrow M$ be a principal fibre bundle with structure group $G$, $F$ a manifold and $\alpha: G\times F\rightarrow F$ a smooth left action.
There is an associated fibre bundle $E\rightarrow ...
1
vote
0
answers
62
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Regularity of a shrunken domain
I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner.
Let $\Omega\subset\Bbb R^d$ be an open bounded (...
13
votes
0
answers
703
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Hilbert 16th problem and dynamical Lefschetz trace formula
I would like to apply the known version of the conjectural formula (11) page !0 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...
4
votes
1
answer
220
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Orientability of moduli space and determinant bundle of ASD operator
Setting
In instanton gauge theory, given a $G$-principal bundle $P\to X^4$, the orientability of the moduli space of ASD connections
$$\mathcal{M}_k = \{A \in L^{2}_{k}(X, \Lambda^1 \otimes\mathrm{...
6
votes
1
answer
373
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Properness of moment map
Suppose that a torus $T$ acts on a non-compact symplectic manifold $M$. Assume that this action is Hamiltonian and that the fixed point set of $T$ is compact. Let $\mu:M\to\mathfrak{t}^{*}$ denote the ...
3
votes
1
answer
275
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Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein
Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein?
I was told that we can use the following method:
Step ...
4
votes
1
answer
518
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Leafwise de Rham cohomology (A true definition of differential forms along leaves)
For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of ...
1
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0
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99
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Inheriting quasiconvexity from convex function after re-parametrisation of space into the Stiefel manifold
Consider a function $f(S): \mathcal{S} \to \mathbb{R}$, where $\mathcal{S}$ is the convex cone of all positive semidefinite complex $M\times M$ Hermitian matrices.
The function $f(\cdot )$ is concave ...
2
votes
1
answer
128
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Closest points of curves on convex surfaces
Let $\Sigma$ be a convex surface with strictly positive curvature in $\mathbb{R}^3$ and $\Gamma \subset \Sigma$ be a closed simple space curve with parameterization $\gamma : \mathbb{S}^1 \rightarrow \...
3
votes
1
answer
865
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Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $?
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.
The characterization of the closed Poincaré dual ...
14
votes
3
answers
2k
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Errata for Bott and Tu's book "Differential Forms in Algebraic Topology"
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Tu is a prequel.
Is there a good list of errata for Bott and Tu available? ...
2
votes
2
answers
561
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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 ...
2
votes
0
answers
126
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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. We quote the ...
2
votes
1
answer
115
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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 ...
1
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0
answers
83
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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 ...
6
votes
1
answer
298
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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$; ...
9
votes
0
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273
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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.
...
1
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0
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114
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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 ...
5
votes
1
answer
469
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Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant
I'm studying properties of the manifold of symmetric positive-definite (SPD) matrices and I've learnt about the following connection to the hyperbolic space [1, Section 2.2], $$\mathcal{P}(n) = \...
3
votes
1
answer
266
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Reduction of the structure group of $\mathbb{R}^n$-fiber bundles to a special subgroup of $\mathrm{Homeo}(\mathbb{R}^n)$
Let $G$ be the group of all self-homeomorphisms $f$ of $\mathbb{R}^n$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n.$$
In other words, $G$ is the group of all equivariant self-...
5
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0
answers
158
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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 ...
5
votes
0
answers
248
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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 ...
1
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0
answers
75
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Foliations with algebraic foliation chart
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 $\...
12
votes
3
answers
3k
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Symmetric and anti-symmetric parts of the covariant derivative of a connection
The following is an excerpt from Sharpe's Differential Geometry - Cartan's Generalization of Klein's Erlangen Program.
Now we come to the question of higher derivatives. As usual in modern
...
12
votes
2
answers
967
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Realizing cohomology classes by submanifolds
In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ...
4
votes
2
answers
294
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Automorphisms of which structure form a Lie groupoid
Given a manifold $M$, the collection of all automorphisms of $M$, denoted by $\text{Aut}(M)$ forms a Lie group.
Do we have similar setting in case of Lie groupoid?
Is there "a structure" whose "...
5
votes
1
answer
383
views
Casson invariant and Euler characteristic
A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference ...
5
votes
1
answer
398
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Poisson summation formula and its implication for the spectrum of the flat torus
I usually ask questions on math.stackexchange but I figure this one is more suited to being asked here. I should preface that I am a complete novice undergraduate, and unlikely to understand answers ...
3
votes
2
answers
577
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When is the action of the gauge group on the space of connections free?
Let $G$ be a compact Lie group. Let $\mathcal{A}$ be the space of connections on the principal trivial $G$-bundle $G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of ...
3
votes
1
answer
139
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Existence of transverse null vector bundle in a degenerate Lorentzian hypersurface
This question is cross-posted at https://math.stackexchange.com/questions/3234895/existence-of-transverse-null-vector-bundle
Let $(M,g)$ be a Lorentzian manifold. Let $\dim(M)=d$. Given a null ...
1
vote
0
answers
106
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Dirac structure
Let $M$ is $m$-dimensional smooth manifold and $\pi: TM\oplus T^{*}M \longrightarrow M $ is a smooth vector bundle. Suppose $L$ is the Dirac structure i.e. it is a subbundle of $(TM \oplus T^{*}M,π)$ ...
13
votes
2
answers
868
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Example of two exotic closed 4-manifolds s.t. SW(X)=0
I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic.
So far in the literature I've only found examples which ...
14
votes
5
answers
1k
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History of the notion of $(G,X)$-structure
I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...
1
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0
answers
49
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Surface swept by a moving line generating constant negative Gauss curvature $K$ and $ \kappa_g$
One end of a flexible curved line in $\mathbb R^3$ moves on the y-axis sweeping out a constant negative Gauss curvature $K$ surface with constant geodesic curvature $\kappa_g$.
Please help find its ...
1
vote
1
answer
201
views
Does fractallity depend on the Riemannian metric?
Edit: According to comment of Andre Henriques we revise the question:
In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...
4
votes
0
answers
204
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Exterior derivative on loop space
Notations:
Let $X$ be a manifold, and denote by $LX := C^\infty(S^1,X)$ its loop space. For a loop $\gamma \in LX$ we can think at the tangent space of $LX$ at the point $\gamma$ as the space of ...
18
votes
2
answers
2k
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Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold
Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta_g$ be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of $\Delta_g$ have finite multiplicity and tend to ...
5
votes
0
answers
51
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Metric structures making the cohomology into a module over a Lie algebra
The cohomology of a closed Kaehler manifold is an $\mathfrak{sl}_2$-module. I think Verbitsky has shown that the cohomology of a closed hyperkaehler manifold is an $\mathfrak{so}_5$-module. For what ...
1
vote
0
answers
129
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Lattices are not solvable in non-compact semisimple Lie groups
I'm trying to prove the following result.
If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...
2
votes
0
answers
252
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Why do unstable manifolds of two close point intersect each other in Baker map?
Let $M$ be $S^1 \times [-1,1]$, $f$ a baker map on $M$ and for $p, q \in M$ consider $W^s_p$ the stable manifold in $p$ (i.e. the set of points whose forward orbit tend to the forward orbit of $p$) ...
6
votes
1
answer
220
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A metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$
This is a cross-post from my question on MSE.
It is well known that
In dimension three a metric has positive sectional curvature if
and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$. where $...
5
votes
1
answer
346
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Long time existence for heat flow in Corlette-Donaldson Theorem
I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14)
https://arxiv.org/pdf/1402.4203.pdf
For completeness, the statement is as follows.
...
1
vote
0
answers
123
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Is this integral zero?
I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation.
Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...
2
votes
0
answers
80
views
Pull backs along rational maps
Let $M^m$ be a compact complex $m$-dimensional manifold and $f: M \dashrightarrow C\mathbb{P}^n$ a rational map (i.e. holomorphic map defined away from a subvariety, $V$, of codimension at least 2). ...
2
votes
0
answers
517
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Euler-Lagrange equations on a differentiable manifold
I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
3
votes
1
answer
775
views
Integration by parts on manifold with corners
Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e.
\begin{align*}
\int_M g(\nabla ...
0
votes
0
answers
62
views
Relationship between the vortex filament equation and the transport equation
Let us consider the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$.
How is the Cauchy problem for the ...
0
votes
1
answer
117
views
Relationship between the vortex filament equation and the cubic Schrödinger equation
How is the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
related to the cubic Schrödinger equation?
Note 1. ...
1
vote
0
answers
70
views
Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...