Metric properties of the space of smooth transformations of a Riemannian manifold (Q1083087)

For two \(C^ 1\)-maps f and g of a Riemannian manifold M into itself a number d(f,g) is defined as \[ d(f,g):=\sup \inf \{l(c)+\| T_ xf- pt(c)\circ T_ xg\| \}, \] where the supremum is formed over all points x of M, the infimum is formed over all \(C^ 1\)-paths c joining g(x) and f(x), l(c) denotes the length of c and pt(c) denotes the parallel transport along c. It is shown that for any fixed map \(f_ 0\) the map d is a metric on the set of maps f for which \(d(f,f_ 0)\) is finite. The analogous result for maps with everywhere invertible derivatives is also proved.