On existence and comparison of conjugate points in Riemannian and Lorentzian geometry
By making use of Jacobi tensors, the existence and comparison of first conjugate points along the geodesics of Riemannian manifolds (or, along the timelike and null geodesics of Lorentzian manifolds) can be studied using the comparison and oscillation theory of linear differential equations. Employing the methods and the results in the comparison and oscillation theory of linear differential equations, a generalized and sharpened Myers theorem, and several generalizations of the Ambrose theorem on existence of conjugate points are obtained. Also, using the comparison theorems for linear differential equations, the first conjugate point along a geodesic of a Riemannian manifold (or, along a timelike or a null geodesic of a Lorentzian manifold) of arbitrary dimension can be compared with the conjugate points along the isotropic geodesics of Riemannian or Lorentzian manifolds (of arbitrary dimensions) by only assuming their Ricci curvatures satisfy certain inequalities. It is shown that, every geodesic (respectively, timelike geodesic) of a 2-dimensional Riemannian manifold (respectively, 2-dimensional Lorentzian manifold) and every null geodesic of a conformally flat Lorentzian manifold of dimension \(n\geq 3\) is isotropic, and therefore, can be used for comparison in the above comparison theorems.
oscillation theory of linear differential equations
scientific article; zbMATH DE number 3937993
scientific article