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Results tagged with dg.differential-geometry
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user 20516
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
18
votes
Quick definition of the tangent space
I think the description of tangent space $T_pM$ you're looking for is one defined entirely locally by "pushing forward" tangent spaces of $\mathbb{R}^n$ via local coordinate charts. ie. the tangent sp …
- 2,274
6
votes
0
answers
256
views
Concrete almost-complex structures on $3 \#CP^2$
The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex structur …
- 2,274
2
votes
1
answer
332
views
density of lagrangian grassmannian in usual grassmannian.
Consider the canonical symplectic structure $(\omega, J)$ on $\mathbb{R}^{2n}$.
(i) What can be said about the density of the lagrangian grassmannian $L$ (i.e. those rank $n$ totally isotropic linear …
- 2,274
4
votes
2
answers
501
views
Linear symmetric spaces are spaces with ''orthogonal complements''?
The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S_n:=SO_n(\mathbb{R}) \backslash SL_n(\mathbb{R})$.
I have only recently been made aware t …
- 2,274
2
votes
2
answers
902
views
How else can we describe the volume of a lagrangian submanifold in a Kahler manifold?
Suppose $(V^{2g}, g, \omega, J)$ is an almost Kahler manifold. ie. $(V,\omega)$ is a symplectic manifold with $\omega$-compatible almost complex structure $J$ ($J$ is a symplectomorphism) and such tha …
- 2,274
17
votes
Accepted
Cotangent bundle lift theorem
[Edited typo 01/16/2022]
Let $\pi:T^*M \to M$ be the canonical projection. Given a diffeomorphism of the base $f:M\to M$, the pullback mapping $f^*:T^*M \to T^*M$ is again a diffeomorphism, and one ha …
- 2,274
1
vote
Linear symmetric spaces are spaces with ''orthogonal complements''?
Finally, i understand the answer! It is Hermann's Convexity Theorem -- and Harish-Chandra's canonical embedding, which proves that EVERY symmetric space (of noncompact type) is conformally equivalent …
- 2,274
6
votes
3
answers
881
views
Reference request: embedded Morse theory
For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further …
- 2,274
0
votes
When is the cut locus a finite tree?
Edited Jan31, 2022: the cut locus of the curve $\gamma$ is a finite tree if the boundary is smooth and if the curve bounds a contractible domain, i.e. $\Omega \approx \{pt\}$. This follows from Blum's …
- 2,274
2
votes
Relation between optimal transport cost and difference between topological invariants?
Applications of OT to Algebraic Topology was the subject of my thesis available here https://github.com/jhmartel/Thesis2019
There remains many interesting questions to solve!
I found the topology of e …
- 2,274