On the structure of locally convex filtrations on complete manifolds

Takao YAMAGUCHI

doi:10.2969/jmsj/04020221

Takao YAMAGUCHI

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ジャーナルフリー

1988 年 40 巻 2 号 p. 221-234

DOI https://doi.org/10.2969/jmsj/04020221

詳細

Published: 1988 Received: 1986/10/20 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) V. Bangert, Riemannsche Mannigfaltigkeiten mit nicht-konstanten konvexer Funktionen, Arch. Math., 31 (1978), 163-170.
2) V. Bangert, Totally convex sets in complete Riemannian manifolds, J. Differential Geom., 16 (1981), 333-345.
3) R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.
4) Yu. D. Burago and V. A. Zalgaller, Convex sets in Riemannian spaces of non-negative curvature, Russian Math. Surveys, 32 (1977), 1-57.
5) H. Busemann and B. B. Phadke, Peakless and monotone functions on G-spaces, Tsu kuba J. Math., 7 (1983), 105-135.
6) J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math., 96 (1972), 413-443.
7) R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: Topological structure, Invent. Math., 63 (1981), 129-157.
8) R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: Differentiable structure, Ann. Sci. École Norm. Sup., 14 (1981), 357-367.
9) R. E. Greene and H. Wu, C^∞ convex functions and manifolds of positive curvature, Acta Math., 137 (1976), 209-245.
10) D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math., 90 (1969), 75-90.
11) M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv., 56 (1981), 179-195.
12) K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math., 106 (1977), 201-211.
13) R. Ichida, Riemannian manifolds with compact boundary, Yokohama Math. J., 29 (1981), 169-177.
14) N. Innami, Totally flat foliations and peakless functions, Arch. Math., 41 (1983), 464-471.
15) D.L. Johnson and L.B. Witt, Totally geodesic foliations, J. Differential Geom., 15 (1980), 225-235.
16) H. Karcher, Schnittort und konvexe Mengen in Vollständigen Riemannschen Mannigfaltigkeiten, Math. Ann., 177 (1968), 105-121.
17) A. Kasue, Ricci curvature, geodesics and geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Japan, 35 (1983), 117-131.
18) V. A. Sharaftdinov, Complete open manifolds of non-negative curvature, Siberian Math. J., 15 (1974), 126-136.
19) R. Walter, On the metric projection onto convex sets in Riemannian spaces, Arch. Math., 25 (1974), 91-98.
20) R. Walter, Some analytical properties of geodesically convex sets, Abh. Math. Sem. Univ. Hamburg, 45 (1976), 263-282.
21) T. Yamaguchi, Locally geodesically quasiconvex functions on complete Riemannian manifolds, Trans. Amer. Math. Soc., 298 (1986), 307-330.
22) T. Yamaguchi, Uniformly locally convex filtrations on complete Riemannian manifolds, Curvature and topology of Riemannian manifolds, Lecture Notes in Math., 1201, Springer, 1986, pp. 303-318.

Right : [1] V. Bangert, Riemannsche Mannigfaltigkeiten mit nicht-konstanten konvexer Funktionen, Arch. Math., 31 (1978), 163-170.
[2] V. Bangert, Totally convex sets in complete Riemannian manifolds, J. Differential Geom., 16 (1981), 333-345.
[3] R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.
[4] Yu. D. Burago and V. A. Zalgaller, Convex sets in Riemannian spaces of non-negative curvature, Russian Math. Surveys, 32 (1977), 1-57.
[5] H. Busemann and B. B. Phadke, Peakless and monotone functions on G-spaces, Tsu kuba J. Math., 7 (1983), 105-135.
[6] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math., 96 (1972), 413-443.
[7] R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: Topological structure, Invent. Math., 63 (1981), 129-157.
[8] R. E. Greene and K. Shiohama, Convex functions on complete noncompact manifolds: Differentiable structure, Ann. Sci. École Norm. Sup., 14 (1981), 357-367.
[9] R. E. Greene and H. Wu, C^∞ convex functions and manifolds of positive curvature, Acta Math., 137 (1976), 209-245.
[10] D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math., 90 (1969), 75-90.
[11] M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv., 56 (1981), 179-195.
[12] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math., 106 (1977), 201-211.
[13] R. Ichida, Riemannian manifolds with compact boundary, Yokohama Math. J., 29 (1981), 169-177.
[14] N. Innami, Totally flat foliations and peakless functions, Arch. Math., 41 (1983), 464-471.
[15] D. L. Johnson and L. B. Witt, Totally geodesic foliations, J. Differential Geom., 15 (1980), 225-235.
[16] H. Karcher, Schnittort und konvexe Mengen in Vollständigen Riemannschen Mannigfaltigkeiten, Math. Ann., 177 (1968), 105-121.
[17] A. Kasue, Ricci curvature, geodesics and geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Japan, 35 (1983), 117-131.
[18] V. A. Sharaftdinov, Complete open manifolds of non-negative curvature, Siberian Math. J., 15 (1974), 126-136.
[19] R. Walter, On the metric projection onto convex sets in Riemannian spaces, Arch. Math., 25 (1974), 91-98.
[20] R. Walter, Some analytical properties of geodesically convex sets, Abh. Math. Sem. Univ. Hamburg, 45 (1976), 263-282.
[21] T. Yamaguchi, Locally geodesically quasiconvex functions on complete Riemannian manifolds, Trans. Amer. Math. Soc., 298 (1986), 307-330.
[22] T. Yamaguchi, Uniformly locally convex filtrations on complete Riemannian manifolds, Curvature and topology of Riemannian manifolds, Lecture Notes in Math., 1201, Springer, 1986, pp. 303-318.

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