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### Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
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### Low bound approximation of a Torus Knot length

Is there a formula for approximating (lower bound) the torus knot length ? The torus knot parameters are (p, q, R, r) where (p,q) are co-primes and R is major axis and r is minor axis of the torus.
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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? ...
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 ...
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 \$\varphi^*...