Volume estimate of submanifolds in compact Riemannian manifolds

Masao MAEDA

doi:10.2969/jmsj/03030533

Masao MAEDA

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JOURNAL FREE ACCESS

1978 Volume 30 Issue 3 Pages 533-551

DOI https://doi.org/10.2969/jmsj/03030533

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Published: 1978 Received: April 13, 1977 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, New York, 1964.
2) J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math., 92 (1970), 61-74.
3) T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math., 83 (1966), 68-73.
4) D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geomerie im Grossen,Springer-Verlag, 1968.
5) N. Grossman, The volume of a totally geodesic hypersurface in a pinched manifold, Paciffic J. Math., 23 (1967), 257-262.
6) N. Grossman, Two applications of the technique of length decreasing variations, Proc. Amer. Math. Soc., 18 (1967), 327-333.
7) C. Heim, Une borne la longueur des géodésiques périodiques d'une variété riemannienecompacte, These, Université Paris, 1971.
8) D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 715-727.
9) F. W. Warner, Extension of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc., 122 (1966), 341-356.

Right : [1] R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, New York, 1964.
[2] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math., 92 (1970), 61-74.
[3] T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math., 83 (1966), 68-73.
[4] D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geomerie im Grossen,Springer-Verlag, 1968.
[5] N. Grossman, The volume of a totally geodesic hypersurface in a pinched manifold, Paciffic J. Math., 23 (1967), 257-262.
[6] N. Grossman, Two applications of the technique of length decreasing variations, Proc. Amer. Math. Soc., 18 (1967), 327-333.
[7] C. Heim, Une borne la longueur des géodésiques périodiques d'une variété riemanniene compacte, These, Université Paris, 1971.
[8] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math., 27 (1974), 715-727.
[9] F. W. Warner, Extension of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc., 122 (1966), 341-356.

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