Behavior of geodesics in foliated manifolds with bundle-like metrics

Shinsuke YOROZU

doi:10.2969/jmsj/03520251

Shinsuke YOROZU

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JOURNAL FREE ACCESS

1983 Volume 35 Issue 2 Pages 251-272

DOI https://doi.org/10.2969/jmsj/03520251

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Published: 1983 Received: December 22, 1981 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, Clairaut submersions, Differential geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, 21-31.
2) P. Dombrowski, Jacobi fields, totally geodesic foliations and geodesic differential forms, Resultate der Math., 1 (1980), 156-194.
3) R.H. Escobales, Jr., Riemannian submersions with totally geodesic fibers, J. Differential Geometry, 10 (1975), 253-276.
4) R. Hermann, On the differential geometry of foliations, Ann. of Math., 72 (1960), 445-457.
5) R. Hermann, Totally geodesic orbits of groups of isometries, Indag. Math., 24 (1962), 291-298.
6) S. Ishihara and M. Konishi, Differential geometry of fibred spaces, Publications of the Study Group of Geometry, 8, Tokyo, 1973.
7) S. Kashiwabara, The structure of a Riemannian manifold admitting a parallel field of one-dimensional tangent vector subspaces, Tohoku Math. J., 11 (1959), 327-350.
8) H. Kitahara, The existence of complete bundle-like metrics, Ann. Sci. Kanazawa Univ., 9 (1972), 37-40.
9) H. Kitahara, The existence of complete bundle-like metrics II, Ann. Sci. Kanazawa Univ., 10 (1973), 51-54.
10) H. Kitahara, The completeness of a Clairaut's foliation, Ann. Sci. Kanazawa Univ., 11 (1974), 37-40.
11) H. Kitahara, On a parametrix form in a certain V-submersion, Lecture Notes in Math., 792, Springer-Verlag, 1980, 264-298.
12) H. Kitahara and S. Yorozu, On the cohomology groups of a manifold with a non-integrable subbundle, Proc. Amer. Math. Soc., 56 (1976), 201-204.
13) H. Kitahara and S. Yorozu, A formula for the normal part of the Laplace-Beltrami operator on the foliated manifold, Pacific J. Math., 69 (1977), 425-432.
14) H. Kitahara and S. Yorozu, On some differential geometric characterizations of a bundle-like metric, Kodai Math. J., 2 (1979), 130-138.
15) S. Kobayashi and K. Nomizu, Foundations of differential geometry, II, Interscience, New York, 1969.
16) R. Maltz, The nullity spaces of the curvature operator, Cahiers Topologie Géom. Différentielle, 8 (1966), 1-20.
17) Y. Muto, On some properties of a fibred Riemannian manifold, Sci. Rep. Yokohama Nat. Univ. Sect. I, 1 (1952), 1-14.
18) H. Nakagawa, Riemannian geometry in the large (in Japanese), Kaigai-boeki, Tokyo, 1977.
19) A.M. Naveira, Variedades foliadas con metrica casifibrada, Collect. Math., 21 (1970), 41-97.
20) B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459-469.
21) B. O'Neill, Submersions and geodesics, Duke Math. J., 34 (1967), 363-373.
22) J.S. Pasternack, Topological obstructions to integrability and the Riemannian geometry of smooth foliations, Thesis, Princeton Univ., 1970.
23) J.S. Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv., 46 (1971), 467-477.
24) B.L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math., 69 (1959), 119-132.
25) B.L. Reinhart, Closed metric foliations, Michigan Math. J., 8 (1961), 7-9.
26) R. Sacksteder, Foliations and pseudogroups, Amer. J. Math., 87 (1965), 79-102.
27) S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J., 10(1958), 338-354.
28) I. Vaisman, Variétés riemanniennes feuilletées, Czechoslovak Math. J., 21 (1971), 46-75.
29) I. Vaisman, Cohomology and differentiable forms, Marcel Dekker, New York, 1973.

Right : [1] R. L. Bishop, Clairaut submersions, Differential geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, 21-31.
[2] P. Dombrowski, Jacobi fields, totally geodesic foliations and geodesic differential forms, Resultate der Math., 1 (1980), 156-194.
[3] R. H. Escobales, Jr., Riemannian submersions with totally geodesic fibers, J. Differential Geometry, 10 (1975), 253-276.
[4] R. Hermann, On the differential geometry of foliations, Ann. of Math., 72 (1960), 445-457.
[5] R. Hermann, Totally geodesic orbits of groups of isometries, Indag. Math., 24 (1962), 291-298.
[6] S. Ishihara and M. Konishi, Differential geometry of fibred spaces, Publications of the Study Group of Geometry, 8, Tokyo, 1973.
[7] S. Kashiwabara, The structure of a Riemannian manifold admitting a parallel field of one-dimensional tangent vector subspaces, Tôhoku Math. J., 11 (1959), 327-350.
[8] H. Kitahara, The existence of complete bundle-like metrics, Ann. Sci. Kanazawa Univ., 9 (1972), 37-40.
[9] H. Kitahara, The existence of complete bundle-like metrics II, Ann. Sci. Kanazawa Univ., 10 (1973), 51-54.
[10] H. Kitahara, The completeness of a Clairaut's foliation, Ann. Sci. Kanazawa Univ., 11 (1974), 37-40.
[11] H. Kitahara, On a parametrix form in a certain V-submersion, Lecture Notes in Math., 792, Springer-Verlag, 1980, 264-298.
[12] H. Kitahara and S. Yorozu, On the cohomology groups of a manifold with a nonintegrable subbundle, Proc. Amer. Math. Soc., 56 (1976), 201-204.
[13] H. Kitahara and S. Yorozu, A formula for the normal part of the Laplace-Beltrami operator on the foliated manifold, Pacific J. Math., 69 (1977), 425-432.
[14] H. Kitahara and S. Yorozu, On some differential geometric characterizations of a bundle-like metric, Kodai Math. J., 2 (1979), 130-138.
[15] S. Kobayashi and K. Nomizu, Foundations of differential geometry, II, Interscience, New York, 1969.
[16] R. Maltz, The nullity spaces of the curvature operator, Cahiers Topologie Géom. Différentielle, 8 (1966), 1-20.
[17] Y. Muto, On some properties of a fibred Riemannian manifold, Sci. Rep. Yokohama Nat. Univ. Sect. I, 1 (1952), 1-14.
[18] H. Nakagawa, Riemannian geometry in the large (in Japanese), Kaigai-boeki, Tokyo, 1977.
[19] A. M. Naveira, Variedades foliadas con metrica casifibrada, Collect. Math., 21 (1970), 41-97.
[20] B. O'Neill, The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459-469.
[21] B. O'Neill, Submersions and geodesics, Duke Math. J., 34 (1967), 363-373.
[22] J. S. Pasternack, Topological obstructions to integrability and the Riemannian geometry of smooth foliations, Thesis, Princeton Univ., 1970.
[23] J. S. Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv., 46 (1971), 467-477.
[24] B. L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math., 69 (1959), 119-132.
[25] B. L. Reinhart, Closed metric foliations, Michigan Math. J., 8 (1961), 7-9.
[26] R. Sacksteder, Foliations and pseudogroups, Amer. J. Math., 87 (1965), 79-102.
[27] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J., 10(1958), 338-354.
[28] I. Vaisman, Variétés riemanniennes feuilletées, Czechoslovak Math. J., 21 (1971), 46-75.
[29] I. Vaisman, Cohomology and differentiable forms, Marcel Dekker, New York, 1973.

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