to arbitrary Riemannian manifolds. Solving the Jacobi equation Let and complete this to get an orthonormal basis at . Parallel transport it to get a basis all along . This gives an orthonormal basis with . The Jacobi field can be written in co-ordinates in terms of this basis as and thus and the Jacobi equation can be rewritten as a system for each . This way we get a linear ordinary differential equation (ODE). Since this ODE has smooth coefficients we have that solutions exist for all and are unique, given and , for all . Examples Consider a geodesic with parallel orthonormal frame , , constructed as above. The vector fields along given by and are Jacobi fields. In Euclidean space (as well as for spaces of constant zero sectional curvature) Jacobi fields are simply those fields linear in . For Riemannian manifolds of constant negative sectional curvature , any Jacobi field is a linear combination of , and , where . For Riemannian manifolds of constant positive sectional curvature , any Jacobi field is a linear combination of , , and , where . The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold. See also Conjugate points Geodesic deviation equation Rauch comparison theorem N-Jacobi field References Manfredo Perdigão do Carmo. Riemannian geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+300 pp. Jeff Cheeger and David G. Ebin. Comparison theorems in Riemannian geometry. Revised reprint of the 1975 original. AMS Chelsea Publishing, Providence, RI, 2008. x+168 pp. Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xvi+468 pp. Barrett O'Neill. Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp. Riemannian geometry Equations