Simplicial structure of the real analytic cut locus
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This note shows how to generalize to arbitrary dimensions the result of S.B. Myers that the cut locus in a real analytic Riemannian surface is triangulable. The basic tool is Hironaka's theory of subanalytic sets. In 1935 and 1936 there appeared in the Duke Mathematical Journal two interesting papers by Sumner B. Myers [3], [4] on the cut locus for a Riemannian manifold. He determined completely the local structure of the cut locus on a two-dimensional real analytic manifold. It turned out that for compact real analytic surfaces the cut locus is a one-dimensional finite simplicial complex. On the other hand in [4] he remarks that "it seems difficult to prove that the locus in the analytic «-dimensional case is homeomorphic to a finite (« l)-dimensional complex". The argument that follows will establish this conjecture. On the one hand we have Hironaka's theory [1], [2] of subanalytic sets which he proved to be triangulable. On the other hand Morse theory gives us a characterization of the cut locus in terms of the behavior of an analytic function on an analytic manifold fibered over the given manifold (the "energy function"). It seemed reasonable then to try to prove the cut locus is subanalytic. This is what the following argument establishes by rather simple reasoning. Let M be a compact analytic Riemannian manifold. Let p E M. If y is a geodesic parametrized by arclength such that y(0) = p then the cut point of (M,p) along y is defined to be the first point y(t0) (t0 > 0) such that for t > í0, y no longer minimizes arclength from/7 to y(t). The set of all such cut points as y varies over all possible geodesies from p is denoted C(p). It is standard that C(p) is characterized by C(p) = {x E M\ either x is the first conjugate point on a length minimizing geodesic starting at p and going through x or there are at least two length minimizing geodesies from/» to x} [6]. Let Sl(M) be the space of piecewise analytic paths starting at p and let E: fí(M) -* R (the energy function) be defined by E(y) = $ \\dy(t)/dtfdt (where we assume now that all paths are parametrized by the unit interval). Choose e > 0 so that whenever d(x, y) < e there is a unique geodesic from x to y of length < e and so that the geodesic depends analytically on the Received by the editors May 12, 1976. AMS (MOS) subject classifications (1970). Primary 53C20; Secondary 32B20, 32C05. © American Mathematical Society 1977