Introductory Geometrical Background

Abstract

Let x ≡ (x ¹,... , x ⁿ) denote local coordinates on a given smooth manifold M ⁿ and let $x \equiv \left( {\frac{{d{x^1}}}{{dt}},...\frac{{d{x^n}}}{{dt}}} \right)$. The system of n 2^nd order ordinary differential equations

$$S:\frac{{{d^2}{x^i}}}{{d{t^2}}} + {g^i}\left( {x,x,i} \right) = 0,i = ...,n$$

has a fully developed differential geometrical characterization discovered by Kosambi (1933, 1935), assisted a bit by E. Cartan (1933) and put into its final elegant form by S. Chern (1939). Whereas, the general theory can be useful (see Section 5.3.1), its most important applications are to be found in the Theory of Sprays (path spaces) which require the above g ⁱ to be homogeneous of degree two in ẋ. In this case, trajectories of S through a given point p ∈ M ⁿ are uniquely specified for each direction, and for any other nearby point q ∈ M ⁿ. Furthermore, there is a canonical parameter t, akin to arc-length in metric geometry so that curves of S become “straight-lines.” There is also the torsion-free spray connection and its deviation curvatures which give information on trajectory stability whose development L. Berwald (1947) credits to Kosambi (ibid.)