ON THE HOLONOMY GROUP OF HYPERSURFACES OF SPACES OF CONSTANT CURVATURE

Ognian Kassabov

Scientific paper ID 1216 : 2015/3

ON THE HOLONOMY GROUP OF HYPERSURFACES OF SPACES OF CONSTANT CURVATURE

Ognian Kassabov

We classify hypersurfaces M^n of manifolds of constant nonzero sectional curvature according their restricted homogeneous holonomy groups. It turns out that outside of the evident cases (restricted holonomy group SO(n) and flat submanifolds) only two cases arise: restricted holonomy group SO(k)×SO(n-k) (when M^n is locally a product of two space forms) and SO(n-1) (when M^n is locally a product of an (n-1) -dimensional space form and a segment).

• Reload article • Open/Download as PDF

пространства с постоянна кривина хиперповърхнини група на холономия.Space of constant curvature hypersurface holonomy groupOgnian Kassabov

BIBLIOGRAPHY

[1] R. Bishop: The holonomy algebra of immersed manifolds of codimension two. Journal of Differ. Geometry, 2(1968), 347-353.

[2] A. Borel and A. Lichnerowicz: Groups d"holonomie des variétés riemanniennes. C. R. Acad. Sci. Paris, 234(1952), 1835-1837.

[3] S. Kobayashi: Holonomy group of hypersurfaces. Nagoya Math. Journal, 10(1956), 9-14.

[4] S. Kobayashi and K. Nomizu: Foundations of differential geometry. Vol. I, John Wiley and Sons, New York, 1963.

[5] R. S. Kulkarni: Equivalence of Kähler manifolds and other equivalence problems. Journal of Differ. Geometry. 9(1974), 401-408.

[6] M. Kurita: On the holonomy group of the conformally flat Riemannian manifold. Nagoya Math. Journal, 9(1955), 161-171.

[7] K. Nomizu and B. Smyth: Differential geometry of complex hypersurfaces II. J. Math. Soc. Japan, 20(1968), 498-521.

[8] G. Vranceanu: Sur les groupes d"holonomie des espaces V_n plongés dans E_(n+p) sans torsion. Revue Roumaine de Math. Pures et Appl., 19(1974), 125-128.