Cut loci, minimizers, and wavefronts in Riemannian manifolds with boundary
The notion of cut loci in manifolds without boundary was discussed by S. Kobayashi, F. W. Warner and A. D. Weinstein [see \textit{S. Kobayashi} and \textit{K. Nomizu}, Foundations of differential geometry. Vol. II (1969; Zbl 0175.485)]. \textit{F.-E. Wolter} [Arch. Math. 32, 92-96 (1979; Zbl 0409.53032); Cut loci in bordered and unbordered Riemannian manifolds, Dissertation, Techn. Univ. Berlin (1985; Zbl 0674.53049)] studied cut locus in Riemannian manifolds with boundary. In fact, Wolter revised the definition of cut locus in Riemannian manifolds with boundary, considered several variants thereof and analyzed the relationship among them. In the present note the authors consider a connected, metrically complete \(C^ \infty\)-Riemannian manifold with boundary, and in the light of the fact that minimizers can bifurcate and merge, there are new features which the authors illustrate by some examples. There may exist open sets each of whose points \(p\) has the property that every point \(q\) sufficiently far from \(p\) has more than one minimizer from \(p\). The authors show that for any \(p\) and almost all \(q\), there is a natural way to choose exactly one minimizer from \(p\) to \(q\). The authors also investigate the ways in which the cut locus of the definition (the cut locus of \(p\) is the closure of the set of points having two minimizers from \(p\) with different terminal velocities) diverges from the classical case. The main result of the paper is as follows: For any given point \(p\), let \(M^*(p)\) consist of the points \(q\) all of whose minimizers from \(p\) have the same terminal velocity vector. Then the complement of \(M^*(p)\) has measure zero, and \(M^*(p)\) is the union of the unique primary minimizers from \(p\) to each of its points. The traces \(pq\) of these minimizers satisfy the branching condition \(pq_ 1 \cap pq_ 2 = pq_ 3\) for some \(q_ 3\).
scientific article; zbMATH DE number 474968
scientific article