A proof of existence of the stable Jacobi tensor (Q1092422)

Consider the (n,n) matrix Jacobi equation (J) \(X''(t)+R(t)X(t)=0\), where R(t) is a given symmetric (n,n) matrix defined for all real numbers t. If X(t) is a solution of (J) that is nonsingular on an interval (a,b), then \(U(t)=X'(t)X^{-1}(t)\) is a solution on (a,b) of the matrix Riccati equation (R) \(U'(t)+U(t)^ 2+R(t)=0\). If one fixes a complete geodesic c in a complete Riemannian manifold M, then one can reduce the equation for Jacobi vector fields on c to the matrix Jacobi equation (J) above. If no two points of c are conjugate with respect to the geodesic flow, then \textit{L. Green} [Mich. Math. J. 5, 31-34 (1958; Zbl 0134.396)] showed by an algebraic argument that there exists a symmetric solution U(t) of (R) that is defined for all t and corresponds to the second fundamental form of the horospheres determined by the geodesic c. Green obtained U(t) from a solution D(t) of (J) that is nonsingular for all t. If the sectional curvature is nonpositive, then the curves \(t\to D(t)x\), \(x\in R^ n\), correspond to the stable Jacobi fields on c, those bounded above in norm for \(t\geq 0\). Green used the existence of the matrix solutions U(t) to prove that the total scalar curvature of a compact Riemannian manifold M without conjugate points is nonpositive and equals zero if and only if M is flat. \textit{A. Freire} and \textit{R. Mañe} [Invent. Math. 69, 375-392 (1982; Zbl 0476.58019)] and \textit{R. Osserman} and \textit{P. Sarnak} [ibid. 77, 455-462 (1984; Zbl 0536.53048)] later used the existence of the solutions U(t) to derive inequalities for the metric entropy of the geodesic flow on a compact negatively curved manifold. In this paper the author uses geometric arguments to give an alternative proof of the fact that if M is complete and simply connected without conjugate points, then each geodesic c admits a symmetric solution U(t) of the corresponding equation (R) that is defined for all t. From the matrices U(t) he then obtains a solution D(t) of (J), defined and nonsingular for all t, which he calls the stable Jacobi tensor for reasons indicated above. The author regards this proof as somewhat simpler than Green's original proof, but Green's proof seems perfectly adequate to this reviewer.