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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 …

[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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …