Averaging of Jacobi fields along geodesics on manifolds of random curvature
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Abstract
The paper considers the Jacobi field along a geodesic on a Riemannian manifold on which the curvature is a stochastic process. The author introduces the concept of linearizing tensor of the Jacobi field on the basis of which a sufficiently universal averaging algorithm is constructed. The equations for higher-order means 〈y p 〉 for p = 2, 3, 4 are deduced. It is shown that these statistical means, as well as the expectation of the Jacobi field, exponentially grow even in the case where the mean value of the curvature vanishes. The growth exponents of higher statistical moments of the Jacobi field obtained analytically with the corresponding exponents obtained from the numerical experiment are compared.
Keywords
Riemannian Manifold Jacobi Equation Close Geodesic Growth Exponent Tensor EquationPreview
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