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### Complex structure of a torus [migrated]

Given the definition of complex structure for a complex manifold:
the real $(1,1)$ type tensor $J_p : T_p M \rightarrow T_p M $ defined by
$$ J_p (\frac{\partial}{\partial x^\mu}) = ...

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### Homogeneous spaces that are homotopy tori

Let $G$ be a compact Lie group, and let $H$ be a closed subgroup such that $G/H$ is homotopy equivalent to a torus. Is it true that $H$ is normal and $G/H$ is isomorphic to a torus as a Lie group?
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### Constructing a vector field with given zeros on a torus

By the Hopf-Poincaré theorem, the sum of the indices of the zeros of a vector field on the d-dimensional torus must equal zero. Given an even number of points $x_i$ on a d-dimensional torus, and ...

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### Special coordinates for periodic metrics

This question is a follow-up to that one.
Given a $\mathbb{Z}^n$-periodic metric $g$ on $\mathbb{R}^n$ (with $n>2$), is it possible to find a periodic diffeomorphism $\varphi$ such that ...