Anosov diffeomorphism

Remarks

The property does not depend on the choice of Riemannian metric, since $M$ is compact. Example: let $A$ be a $2 \times 2$ integer matrix with $\det A = \pm 1$ and no eigenvalue of modulus $1$, and let $f_A$ be the induced map of the torus $\R^2/\Z^2$. Then $Df_A = A$ at every point, and the eigenlines of $A$ give $E^s$ and $E^u$ with $C = 1$ after a linear change of metric. For $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ the eigenvalues are $(3 \pm \sqrt5)/2$, which gives Arnold's cat map. Non-example: for $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ the map $f_A$ is a diffeomorphism of the torus, but both eigenvalues equal $1$, so no direction is contracted and the condition fails. All known Anosov diffeomorphisms are, up to topological conjugacy, hyperbolic automorphisms of tori, nilmanifolds or infranilmanifolds; whether this is always the case is an open question.