Characterizations of complex space forms by means of geodesic spheres and tubes

J. Gillard

Colloquium Mathematicae (1996)

  • Volume: 71, Issue: 2, page 253-262
  • ISSN: 0010-1354

Abstract

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We prove that a connected complex space form ( M n ,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition R ˜ X Y · ϱ ˜ = 0 and by the semi-parallel condition R ˜ X Y · σ = 0 , considering special choices of tangent vectors X , Y to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where R ˜ , ϱ ˜ and σ denote the Riemann curvature tensor, the corresponding Ricci tensor of type (0,2) and the second fundamental form of the spheres or tubes and where R ˜ X Y acts as a derivation.

How to cite

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Gillard, J.. "Characterizations of complex space forms by means of geodesic spheres and tubes." Colloquium Mathematicae 71.2 (1996): 253-262. <http://eudml.org/doc/210439>.

@article{Gillard1996,
abstract = {We prove that a connected complex space form ($M^n$,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition $\tilde\{R\}_\{XY\}·\tilde\{ϱ\}=0$ and by the semi-parallel condition $\tilde\{R\}_\{XY\}·σ=0$, considering special choices of tangent vectors $X,Y$ to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where $\tilde\{R\}$, $\tilde\{ϱ\}$ and $σ$ denote the Riemann curvature tensor, the corresponding Ricci tensor of type (0,2) and the second fundamental form of the spheres or tubes and where $\tilde\{R\}_\{XY\}$ acts as a derivation.},
author = {Gillard, J.},
journal = {Colloquium Mathematicae},
keywords = {complex space forms; geodesic spheres; tubular hypersurfaces about geodesics},
language = {eng},
number = {2},
pages = {253-262},
title = {Characterizations of complex space forms by means of geodesic spheres and tubes},
url = {http://eudml.org/doc/210439},
volume = {71},
year = {1996},
}

TY - JOUR
AU - Gillard, J.
TI - Characterizations of complex space forms by means of geodesic spheres and tubes
JO - Colloquium Mathematicae
PY - 1996
VL - 71
IS - 2
SP - 253
EP - 262
AB - We prove that a connected complex space form ($M^n$,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition $\tilde{R}_{XY}·\tilde{ϱ}=0$ and by the semi-parallel condition $\tilde{R}_{XY}·σ=0$, considering special choices of tangent vectors $X,Y$ to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where $\tilde{R}$, $\tilde{ϱ}$ and $σ$ denote the Riemann curvature tensor, the corresponding Ricci tensor of type (0,2) and the second fundamental form of the spheres or tubes and where $\tilde{R}_{XY}$ acts as a derivation.
LA - eng
KW - complex space forms; geodesic spheres; tubular hypersurfaces about geodesics
UR - http://eudml.org/doc/210439
ER -

References

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  1. [1] E. Boeckx, J. Gillard and L. Vanhecke, Semi-symmetric and semi-parallel geodesic spheres and tubes, Indian J. Pure Appl. Math., to appear. Zbl0866.53037
  2. [2] B. Y. Chen and L. Vanhecke, Differential geometry of geodesic spheres, J. Reine Angew. Math. 325 (1981), 28-67. 
  3. [3] M. Djorić and L. Vanhecke, Almost Hermitian geometry, geodesic spheres and symmetries, Math. J. Okayama Univ. 32 (1990), 187-206. Zbl0735.53049
  4. [4] L. Gheysens and L. Vanhecke, Total scalar curvature of tubes about curves, Math. Nachr. 103 (1981), 177-197. Zbl0505.53017
  5. [5] A. Gray, Tubes, Addison-Wesley, Reading, 1989. 
  6. [6] A. Gray and L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math. 142 (1979), 157-198. Zbl0428.53017
  7. [7] A. Gray and L. Vanhecke, The volumes of tubes about curves in a Riemannian manifold, Proc. London Math. Soc. 44 (1982), 215-243. Zbl0491.53035
  8. [8] S. Tanno, Constancy of holomorphic sectional curvature in almost Hermitian manifolds, Kōdai Math. Sem. Rep. 25 (1973), 190-201. Zbl0263.53019
  9. [9] L. Vanhecke, Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari, Supplemento al Vol. 58 (1988), 73-176. 
  10. [10] L. Vanhecke and T. J. Willmore, Interaction of tubes and spheres, Math. Ann. 263 (1983), 31-42. Zbl0491.53034
  11. [11] K. Yano and M. Kon, Structures on Manifolds, Ser. in Pure Math. 3, World Sci., Singapore, 1984. 

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