Metric invariants and space structures

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Distance function d_p to a point p of a compact Riemannian manifold is an important geometric function that is also related to the manifold structure. However, distance function d_p admits a point where d_p is not differentiable, and such a point is contained in the cut locus C_p of p. It was known that the notion of critical points may be introduced as in usual Morse theory and d_p is not differentiable at critical points. In the case where there are no critical points, Morse theory (the isotopy lemma) has been developed as in smooth case that played an important role in applications.Now to develop Morse theory for distance functions including critical points, one should first define the notion of the index at a critical point. Under the support of the present Grant-in-Aid, T. Sakai, head investigator of this research program, carried out this with J. Itoh when Riemannian metric satisfies some natural non-degeneracy conditions, and developed Morse theory for distance functions from a … More direct geometric view point (has been published in a Journal). There it is important to show that the cut locus Cp carries a nice structure (Whitney stratification), and the index of a critical point q is defined in terms of the number of minimal geodesics joining p, q and the usual index at q of the restriction of 4 to the stratum containing q that is a smooth function with a critical point q. I hope to continue to study how generic is the conditions among all Riemannian metrics on a given manifold.With respect to distance functions, K. Kiyohara determined the explicit structure of the cut locus and the conjugate locus of any point in ellipsoids (and some Liouville manifolds) from a view point of integrable geodesic flow with J. Itoh, and A. Katsuda studied the inverse problem of the Neumann boundary value problem, namely how to reconstruct the inner Riemannian metric from the distance function to the boundary of a Riemannian manifold with boundary.As for geometric inequalities (isosystolic inequality, isodiametric inequality) and the first eigenvalue estimate of the Laplacian of a compact Riemannian manifolds with non-negative Ricci curvature and its perturbation, we could not make substantial progress in this period, but Kiyohara obtained a result concerning an inequality of geometric invariants of Alexandrov spaces. I hope to continue to carry the program. Y. Mori investigated algorithm of quantum computer and supported Sakai in computer aid. Less