On conjugate and focal points in semi-Riemannian geometry
The author proves several theorems on the existence and comparison of conjugate and focal points in semi-Riemannian manifolds. Rather than using the index form, the proofs use oscillation and comparison theorems for differential equations applied to the Raychaudhuri equation (Riccati equation). The Riemannian and non-spacelike Lorentzian cases were treated previously by the author [Math. Ann. 276, 67--79 (1986; Zbl 0585.53037)]. A conjugate point is said to be nondegenerate if the span of the velocity vectors of vanishing Jacobi fields at the point is a nondegenerate vector space. It is shown that nondegenerate conjugate points are isolated and that a conjugate point is isolated if the trace of the square shear can be extended to a continuous function on a neighborhood of the point. The author proves several theorems on the relation of Ricci and sectional curvature to the existence of conjugate points and focal points, including Myer's theorem for both conjugate points and focal points in a semi-Riemannian manifold. A geodesic is said to be codefinite if the index of the induced metric on the nondegenerate orthogonal bundle \(\gamma^{\perp}\) is zero or equal to the rank of \(\gamma^{\perp}\). A geodesic is said to be isotropic if the curvature endomorphism defined on \(\gamma^{\perp}\) is a multiple of the identity at each point of the geodesic. The author obtains both conjugate point and focal point comparison theorems between codefinite and isotropic geodesics. It is shown that the isotropic condition yields a vanishing shear tensor along a geodesic. Also, a null geodesic is isotropic if the projection of the Weyl tensor to the nondegenerate orthogonal bundle is identically zero.
scientific article; zbMATH DE number 4076009
scientific article