Curvature and the Variation Formula

Abstract

This chapter is a direct continuation of Chapter IX, but in a new context, the variation formula. Given a family of curve {α _t }, their lengths L(α _t ) defines a function, and we are interested in the singular points of this function on the space of curves especially the relative minima and the second derivative test. We do not formalize the infinite dimensional space of curves but work simply with families. We shall see that the Riemann tensor plays an essential role in the expression for the second derivative, which allows us to go futher than we did in Chapter IX, and especially in proving the converse of Theorem 3.6, for which we have to deal with positive curvature. The variation formula will allow us to estimate growths of Jacobi lifts more generally than in Chapter IX.