Geometry of weakly symmetric spaces

Jürgen BERNDT; Lieven VANHECKE

doi:10.2969/jmsj/04840745

Jürgen BERNDT, Lieven VANHECKE

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1996 Volume 48 Issue 4 Pages 745-760

DOI https://doi.org/10.2969/jmsj/04840745

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Published: 1996 Received: November 18, 1994 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) J. Berndt, Real hypersurfaces with constant principal curvatures in complex space forms, Geometry and Topology of Submanifolds II, Proc. Conf. Avignon/France 1988, world Scientific, Singapore, 1990, pp. 10-19.
2) J. Berndt, O. Kowalski and L. Vanhecke, Geodesics on weakly symmetric spaces, preprint, 1995.
3) J. Berndt, F. Prüfer and L. Vanhecke, Symmetric-like Riemannian manifolds and geodesic symmetries, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 265-282.
4) J. Berndt, F. Prüfer and L. Vanhecke, Geodesic spheres and two-point homogeneous spaces, Israel J. Math., 93 (1996), 373-385.
5) J. Berndt, F. Tricerri and L. Vanhecke, Geometry of generalized Heisenberg groups and their Damek-Ricci harmonic extensions, C.R. Acad. Sci. Paris Sér. I, 318 (1994), 471-476.
6) J. Berndt and L. Vanhecke, Two natural generalizations of locally symmetric spaces, Diff. Geom. Appl., 2 (1992), 57-80.
7) J. Berndt and L. Vanhecke, Naturally reductive Riemannian homogeneous spaces and real hypersurfaces in complex and quaternionic space forms, Differential Geometry and Its Applications, Proc. Conf. Opava (Czechoslovakia) 1992, Silesian University at Opava, 1993, pp. 353-364.
8) L. Bieszk, On natural reductivity of five-dimensional commutative spaces, Note Mat., 8 (1988), 13-43.
9) J.P. Bourguugnon and H. Karcher, Curvature operators: pinching estimates and geometric examples, Ann. Sci. École Norm. Sup., 11 (1978), 71-92.
10) B.Y. Chen and L. Vanhecke, Isometric, holomorphic and symplectic reflections, Geom. Dedicata, 29 (1989), 259-277.
11) M. Cowling, A.H. Dooley, A. Korányi and F. Ricci, H-type groups and Iwasawa decompositions, Adv, in Math., 87 (1991), 1-41.
12) J.E. D'Atri, Geodesic spheres and symmetries in naturally reductive spaces, Michigan Math. J., 22 (1975), 71-76.
13) S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.
14) G. Jensen, Embeddings of Stiefel manifolds into Grassmannians, Duke Math. J., 42 (1975), 397-407.
15) S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I, Interscience Publishers, New York, 1963.
16) S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. II, Interscience Publishers, New York, 1969.
17) O. Kowalski, Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale, (Settembre 1983), 131-158.
18) O. Kowalski and L. Vanhecke, Four-dimensional naturally reductive homogeneous spaces, Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale, (Settembre 1983), 223-232.
19) O. Kowalski and L. Vanhecke, Two-point functions on Riemannian manifolds, Ann. Global Anal. Geom., 3 (1985), 95-119.
20) O. Kowalski and L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. (7), 5-B (1991), 189-246.
21) A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47-87.
22) Z.I. Szabó, Spectral geometry for operator families on Riemannian manifolds, Proc. Sympos. Pure Math., 54 (1993), 615-665.
23) J. Tits, Sur certaines classes d'espaces homogènes de grouper de Lie, Mem. Acad. Roy. Belg., 29, 1955.
24) L. Vanhecke, Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari, Supplemento al, vol. 58 (1988), 73-176.
25) H.C. Wang, Two-point homogeneous spaces, Ann. Math., 55 (1952), 177-191.
26) J.A. Wolf, Elliptic spaces in Grassmann manifolds, Illinois J. Math., 7 (1963), 447-462.
27) W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann., 259 (1982), 351-358.

Right : [1] J. Berndt, Real hypersurfaces with constant principal curvatures in complex space forms, Geometry and Topology of Submanifolds II, Proc. Conf. Avignon/France 1988, world Scientific, Singapore, 1990, pp. 10-19.
[2] J. Berndt, O. Kowalski and L. Vanhecke, Geodesics on weakly symmetric spaces, preprint, 1995.
[3] J. Berndt, F. Prüfer and L. Vanhecke, Symmetric-like Riemannian manifolds and geodesic symmetries, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 265-282.
[4] J. Berndt, F. Prüfer and L. Vanhecke, Geodesic spheres and two-point homogeneous spaces, Israel J. Math., 93 (1996), 373-385.
[5] J. Berndt, F. Tricerri and L. Vanhecke, Geometry of generalized Heisenberg groups and their Damek-Ricci harmonic extensions, C. R. Acad. Sci. Paris Sér. I, 318 (1994), 471-476.
[6] J. Berndt and L. Vanhecke, Two natural generalizations of locally symmetric spaces, Diff. Geom. Appl., 2 (1992), 57-80.
[7] J. Berndt and L. Vanhecke, Naturally reductive Riemannian homogeneous spaces and real hypersurfaces in complex and quaternionic space forms, Differential Geometry and Its Applications, Proc. Conf. Opava (Czechoslovakia) 1992, Silesian University at Opava, 1993, pp. 353-364.
[8] L. Bieszk, On natural reductivity of five-dimensional commutative spaces, Note Mat., 8 (1988), 13-43.
[9] J. P. Bourguugnon and H. Karcher, Curvature operators: pinching estimates and geometric examples, Ann. Sci. École Norm. Sup., 11 (1978), 71-92.
[10] B. Y. Chen and L. Vanhecke, Isometric, holomorphic and symplectic reflections, Geom. Dedicata, 29 (1989), 259-277.
[11] M. Cowling, A. H. Dooley, A. Korányi and F. Ricci, H-type groups and Iwasawa decompositions, Adv, in Math., 87 (1991), 1-41.
[12] J. E. D'Atri, Geodesic spheres and symmetries in naturally reductive spaces, Michigan Math. J., 22 (1975), 71-76.
[13] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.
[14] G. Jensen, Embeddings of Stiefel manifolds into Grassmannians, Duke Math. J., 42 (1975), 397-407.
[15] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I, Interscience Publishers, New York, 1963.
[16] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. II, Interscience Publishers, New York, 1969.
[17] O. Kowalski, Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale, (Settembre 1983), 131-158.
[18] O. Kowalski and L. Vanhecke, Four-dimensional naturally reductive homogeneous spaces, Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale, (Settembre 1983), 223-232.
[19] O. Kowalski and L. Vanhecke, Two-point functions on Riemannian manifolds, Ann. Global Anal. Geom., 3 (1985), 95-119.
[20] O. Kowalski and L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. (7), 5-B (1991), 189-246.
[21] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47-87.
[22] Z. I. Szabó, Spectral geometry for operator families on Riemannian manifolds, Proc. Sympos. Pure Math., 54 (1993), 615-665.
[23] J. Tits, Sur certaines classes d'espaces homogènes de grouper de Lie, Mem. Acad. Roy. Belg., 29, 1955.
[24] L. Vanhecke, Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari, Supplemento al, vol. 58 (1988), 73-176.
[25] H. C. Wang, Two-point homogeneous spaces, Ann. Math., 55 (1952), 177-191.
[26] J. A. Wolf, Elliptic spaces in Grassmann manifolds, Illinois J. Math., 7 (1963), 447-462.
[27] W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann., 259 (1982), 351-358.

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