On diffeomorphisms of a hyperbolic manifold (Q2752882)

This is a short note about diffeomorphisms of compact hyperbolic manifolds. Assume that we have such a diffeomorphism \(f\) whose displacement at every point is less than the injectivity radius. Thus one can define a vector field \(W\) by taking the inverse of the exponential: namely at each point \(p\) the vector \(W_p\) is defined by the rule \(\exp(W_p)=f(p)\). The theorem states that if the derivative of \(W_p\) has norm less than one, then \(f\) can be isotoped to the identity by a family of diffeomorphisms. In fact the isotopy is the natural one: for each time \(t\in [0,1]\) take the diffeomorphism that maps \(p\) to \(\exp(t W_p)\). The author sketches a proof based on Jacobi fields, that this map is a diffeomorphisms.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00049].