IR is called uniformly convex (concave) if Q is a convex open set and if it is a C 2 -function satisfying Ixx > 0 ([xx< 0). Note that uniform convexity implies the strict convexity condition I(Äx if x, Z E Q and x #- z.

On the other hand we ha ve (5) h*IC H = 11i dcpi - H(h) dx = (11icpif - h*H) dx + 11iCP~ dca.. Then we infer from (2)-(5) Lemma 1. a(x, c) are given by (7) We call (6) a Cauchy representation of h*IC H in terms of the eigentime S. By taking the exterior differential of h*IC H we obtain 3 In German: "Eigenzeit". 2. Hamiltonian Flows and Their Eigentime Functions 35 Lemma 2. ~ = uX [1/; .. + HAh)]

5) Unfortunately, there is no unanimously accepted terminology in the literat ure. Therefore we shall not stick to our nomenclature very rigorously but we shall use different names in different situations. Presently we want to view graph u = {(x, z): z = u(x), x E Q} as a nonparametric surface in IR" x IR given by a mapping u: Q -+ IR N , Q c IR". Hence x = (x I, ... ,x") are not merely parameters but geometric coordinates enjoying the same rights as z = (Zl, ... ,ZN). The geometric object is an n-dimensional surface Y = graph u of codimension N sitting in IR" x IR N ; therefore the configuration space C(f is in this situation thought to be the x, z-space.