Simplicial structure of the real analytic cut locus

Author:
Michael A. Buchner

Journal:
Proc. Amer. Math. Soc. **64** (1977), 118-121

MSC:
Primary 53C20; Secondary 32B20, 57D70

DOI:
https://doi.org/10.1090/S0002-9939-1977-0474133-5

MathSciNet review:
0474133

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Abstract: 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.

**[1]**Heisuke Hironaka,*Subanalytic sets*, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453–493. MR**0377101****[2]**Heisuke Hironaka,*Triangulations of algebraic sets*, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 165–185. MR**0374131****[3]**Sumner Byron Myers,*Connections between differential geometry and topology. I. Simply connected surfaces*, Duke Math. J.**1**(1935), no. 3, 376–391. MR**1545884**, https://doi.org/10.1215/S0012-7094-35-00126-0**[4]**Sumner Byron Myers,*Connections between differential geometry and topology II. Closed surfaces*, Duke Math. J.**2**(1936), no. 1, 95–102. MR**1545908**, https://doi.org/10.1215/S0012-7094-36-00208-9**[5]**J. Milnor,*Morse theory*, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR**0163331****[6]**M. Buchner,*Stability of the cut locus in dimensions*(to appear).

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0474133-5

Article copyright:
© Copyright 1977
American Mathematical Society