ON A COMPUTATIONAL METHOD FOR THE 2ND FUNDAMENTAL TENSOR AND ITS APPLICATION TO BIFURCATION PROBLEMS

An algorithm is presented for the computation of the second fundamental tensor V of a Riemannian submanifold M of Rn. 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 d-dimensional manifold M is defined implicitly as the zero set of a submersion F on Rn. In this case, the principal cost of the algorithm for computing V(p) at a given point p∈M involves only the decomposition of the Jacobian DF(p) of F at p and the projection of d(d+1) neighboring points onto M by means of a local iterative process using DF(p). Several numerical examples are given which show the efficiency and dependability of the method. © 1990 Springer-Verlag.