By P. Ciarlet

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**Additional resources for An Intro. to Differential Geometry With Applns to Elasticity**

**Sample text**

If X, Yare f-martingales of bounded dilatation on N with Xo and Yo not equal, then the probability that they never meet is positive. Proof. This follows the now-familiar lines of Ito analysis of a distance, namely distance(X, Y). A comparison argument shows it is greater than a Brownian motion of constant positive drift. 8). One geometric implication is immediate. THEOREM 14. If M has the BCP and N is as in the previous theorem, then every harmonic map of bounded dilatation F:M~N is constant. Proof.

The analysis of ar is deeper and depends on the notion of curvature. This is the crux of stochastic differential geometry; all we can give here is a brief intuitive discussion and a reference to Cheeger and Ebin (1975) 'or Milnor (1963) for rigorous treatm~nt. 46 WILFRID S. 1. CURVATURE Consider two particles moving at constant speed along geodesics originating from a common point but issuing at slightly different angles. To the first order the distance between the particles increases at a linear rate for small times.

The fibre is isometric to Euclidean space V of the same dimension mas M; let Ox(M) be the collection of all isometries from V to TxM and OeM) be the union of all fibres Ox(M) as x runs through M, the orthonormal frame bundle. ) Let 7T be the obvious projection 7T: OeM) ~ M. We require a semimartingale 5 with 7T(5) = X. 1) (note that 5- 1 is also a semimartingale on a suitable manifold so that the Stratonovich differential equation above makes sense). The crux of the matter is to see how to define 5.