On a Computational Method for the Second Fundamental Tensor and Its Application to Bifurcation Problems.
PITTSBURGH UNIV PA INST FOR COMPUTATIONAL MATHEMATICS AND APPLICATIONS
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An algorithm is presented for the computation of the second fundamental tensor V of a Riemannian sub-manifold M of R superscript n. From V the Riemann curvature tensor of M is easily obtained. Moreover, V has a close relation to the second derivative of certain functionals on M which, in turn, provides a powerful new tool for the computational determination of multiple bifurcation directions. Frequently, in applications, the manifold M is defined implicitly as the zero set of a submersion F on R superscript n. In this case, the principal cost of algorithm for computing V p at a given point p an element of M involves only the decomposition of the Jacobian DF p of F at p and the projection of dd1 neighboring points onto M by means of a local iterative process using DFp. Several numerical examples are given which show the efficiency and dependability of the method. jhd
- Theoretical Mathematics