On the distribution of conjugate points (Q582601)

A new criterion for the existence of conjugate points is obtained by use of Morse theory. Theorem: Let c: [0,L]\(\to M\) be a unit speed geodesic on a Riemannian manifold of dimension n. If \[ \int^{L}_{0}Ric(c'(t))dt\geq \pi (n-1)^{1/2}\sqrt{\max_{t\in [0,L]}(0,Ric(c'(t)))}, \] and \(Ric(c')\) is not identically zero, then c(0) has a conjugate point c(T) along c for some T in (0,L]. Furthermore, if the smallest such T is L, then \(K(\sigma)=\pi^ 2/L^ 2\) for all tangent two planes \(\sigma\) containing a tangent vector to c. This criterion, with the help of Birkhoff's ergodic theorem, gives the main result. Theorem: Let M be a complete Riemannian manifold of dimension n with a finite volume and Ricci curvature bounded above. Then \[ \int_{M}Scal \leq \frac{\pi (n-1)^{1/2}n}{vol(S^{n-1},can)}\sup (0,Ric)\int_{SM}\psi. \] In the theorem, SM is the unit tangent bundle, Ric is the Ricci curvature, and \(\psi: SM\to [0,\infty]\) is defined by \(\psi (v)=\liminf_{T\to \infty}(1/T)\times\) (the number of points conjugate to \(c_ v(0)\) along \(c_ v|_{[0,T]}\), where \(c_ v(t)=\exp (tv))\).