All Questions
46 questions
1
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0
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152
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Invariants associated to a principal bundle whose total space is a symplectic manifold acted symplectically by group structure
The following question - proposal came to my mind about 4 years ago but I did not find any solution to this question and did not find any answer via e-personal comunication with some ...
- 356
2
votes
1
answer
326
views
Sectional curvature and injectivity radius of natural metric in cotangent bundles
In the following paper by Cielibak, Ginzburg and Kerman (arXiv link, Comm. Math. Helv. 2004 DOI link) they claim in page $3$ that the natural metric $\tilde g$ on $T^*M$ the sectional curvature is ...
user174565
1
vote
0
answers
210
views
Proving that a twisted cotangent bundle is geometrically bounded
In Cieliebak, Ginzburg and Kerman's paper Symplectic homology and periodic orbits near symplectic submanifolds, the authors claim and give a proof that a twisted cotangent bundle will be geometrically ...
- 791
2
votes
0
answers
70
views
Compatible almost complex structures such that the associated riemannian metric has positive injectivity radius
Let $M$ be a compact manifold, consider $\omega$ the canonical symplectic form in $T^*M$ and $\hat J$ the canonical almost complex structure coming from the Sasaki metric.
Let $\mathcal{J}$ be the set ...
- 791
5
votes
1
answer
564
views
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 $...
- 356
4
votes
0
answers
104
views
Non-isomorphic compact Kähler manifolds not containing submanifolds biholomorphic to their conjugates
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\nu:M\to N$ ...
user164740
2
votes
1
answer
191
views
Non-symplectomorphic isometric compact Kähler manifolds
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\phi:M\to N$...
user164740
13
votes
1
answer
493
views
Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\nu:M\to N$ ...
user164740
3
votes
1
answer
366
views
When is the exterior derivation $d$ a Lie algebra morphism?
In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We ...
- 356
2
votes
1
answer
158
views
A metric naturally arise from the Euclidean symplectic structure?
For $n>1$ let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$.
We define the following distribution $D$ on $\mathbb{...
- 356
16
votes
1
answer
1k
views
Is there an area-preserving diffeomorphism of the disk which is nowhere conformal?
This question is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk.
Does there exist a smooth area-preserving diffeomorphism $f:D \to D$ that does not have conformal points?
I ...
- 6,741
8
votes
2
answers
695
views
Kronheimer's results on ALE spaces as hyperkahler quotients
Background: In his two papers from late 80s Kronheimer proved that any 4-dimensional ALE space is given by a hyperkahler quotient, say $X_{{\zeta_\mathbb{R}},{\zeta_\mathbb{C}}}(Q)$ where Q is a ...
- 1,677
7
votes
2
answers
944
views
Volume of manifolds embedded in $\mathbb{R}^n$
Let $N$ be a closed, connected, oriented hypersurface of $\mathbb{R}^n$. Such a manifold inherits a volume form from the usual volume from on $\mathbb{R}^n$ and has an associated volume given by ...
- 979
5
votes
2
answers
388
views
Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces
I apologize in advance if this question has an obvious answer.
Let $(M,g)$ be a Riemannian manifold.
Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...
- 356
4
votes
0
answers
119
views
Existence of harmonic symplectic structure on symplectic Riemannian manifold
This post is an expanded version of this MSE post.
Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric.
Is there a symplectic structure $\...
- 356
9
votes
1
answer
677
views
Does every manifold admit a Lagrangian Riemannian metric?
Let $(M,g)$ be a Riemannian manifold. The $LC$ connection associated to the metric gives an $n$ dimensional distribution $D$ for $TM$. Let $\omega$ be the symplectic structure of $TM$ which is ...
- 356
8
votes
1
answer
256
views
Do all symmetries of a Kähler quotient come from the original space?
For a Kähler manifold $M$, let $\operatorname{Iso}_{\mathbb{C}}(M)$ denote the group of holomorphic isometries.
Suppose that $K$ is a compact subgroup of $\operatorname{Iso}_{\mathbb{C}}(M)$ and ...
- 81
28
votes
6
answers
3k
views
Why is there no symplectic version of spectral geometry?
First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta_g:C^\infty(M)\to C^\infty(M)$ is defined as
$$
\Delta_g=\mathrm{div}_g\circ\mathrm{grad}_g,
$$
where the ...
- 1,942
2
votes
1
answer
390
views
A Lagrangian connection and its algebraic interpretation
Assume that $M$ is a $n$ dimensional Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold.
A Lagrangian connection $D$ on $M$ is a $n$ dimensional distribution for $TM$ such ...
- 356
4
votes
0
answers
191
views
A quantity associated with a Riemannian surface
Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ ...
- 356
0
votes
2
answers
553
views
A relation between gradient vector field and Hamiltonian vector field
Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.
Is there a Riemannian ...
- 356
15
votes
2
answers
829
views
Darboux-like theorems
Related to Kahler version of Darboux's Theorem
I know a few theorems that feel like Darboux's theorem. By that, I mean some kind of geometry based around a "pointwise" condition and the existence ...
- 4,991
2
votes
1
answer
590
views
Ricci form is closed?
Let $(M,g,J)$ be an almost Kähler manifold and let $\rho$ denote its Ricci form
$$
\rho(X,Y) = \operatorname{ric}^{\mbox{c}}(JX,Y)
$$
where $\operatorname{ric}^{\mbox{c}}$ is the $J$-invariant part of ...
1
vote
0
answers
172
views
The Lie algebra of Harmonic functions
Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...
- 356
2
votes
1
answer
121
views
coisotropic action on $TS^{2n+1}$
Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of $SU(...
- 501
2
votes
0
answers
144
views
Flow on invariant Lagrangian tori
The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...
- 13.5k
5
votes
1
answer
420
views
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 $...
- 501
2
votes
1
answer
244
views
Symplectic reduction: from indefinite signature to Riemannian signature
Let $(M,g,\omega)$ be a $d$-dimensional manifold equipped with a metric $g$ of signature $(t,s)$, $d = t+s$, and a symplectic form $\omega$. Let us assume that a Lie group $G\subset Isometries(M,g)$ ...
- 2,816
1
vote
1
answer
215
views
Relation between volume of reduced space and phase space
Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is $$...
user21574
5
votes
1
answer
1k
views
What is the geometric interpretation of this quantity?
Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic
structure on the tangent bundle, $(TM, \...
- 356
6
votes
1
answer
868
views
Shortest geodesic loop vs. shortest periodic geodesic
Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic?
For example, is this true for small ...
- 13.5k
1
vote
1
answer
510
views
The space of generalized complex structures in sense of N.Hitchin is contractible?
Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...
user21574
1
vote
1
answer
227
views
choices of connection in prequantization
In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...
user21574
1
vote
1
answer
177
views
pre-symplectic and foliation and its trajectories
Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?
user21574
4
votes
1
answer
467
views
Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold
Hallo,
I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ,...
- 53
4
votes
2
answers
626
views
Uniqueness of Kähler form with same volume
Hallo,
Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure $...
- 339
1
vote
1
answer
218
views
Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold
Hallo,
Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...
- 339
8
votes
1
answer
458
views
Different complexifications of a real analytic Riemannian manifold
I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well known fact that in a neighbourhood $U$ of the zero ...
- 101
3
votes
1
answer
388
views
Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold
Hallo,
It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...
- 339
0
votes
1
answer
339
views
Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
9
votes
2
answers
2k
views
$J$-holomorphic curve as a minimal surface
The following is a part of the proof of Gromov nonsqueezing theorem.
The existence of a $J$-holomorphic curve gives an upper bound for the radius of a symplectically embedded ball.
Let $\psi: B(r) \...
- 1,398
1
vote
1
answer
398
views
Holonomy group of cotangent bundle
Is the holonomy group of the cotangent bundle, of a compact riemannian manifold, with respect to te standard symplectic structure equal to $SU(n)$, where $n$ is the dimension of the riemannian ...
- 13
18
votes
5
answers
2k
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Is there a coordinate-free proof of the hamiltonian character of the geodesic flow?
I do not know if this question is appropriate for this site, but I posted here without having answers, so now I make this attempt.
Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the ...
- 4,306
8
votes
1
answer
2k
views
Calculating the geodesic equation for a particular set of phase-space coordinates
Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ ...
- 8,512
3
votes
2
answers
657
views
Reference for Almost-Kahler geometry
Is there any comprehensive reference for Almos-Kahler geometry or more generally to Almost- Hermitian geometry ?
- 1,236
24
votes
5
answers
2k
views
When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold?
Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ...
- 29.2k