Abstract
The orbits of Lie groups acting in Euclidean spaces by isometries are extrinsically symmetric iff they are parallel, i.e. satisfy % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0. Submanifolds characterized by the integrability condition \-R · h = 0 of this system % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0 are called semi-parallel; they are the second order envelopes of the symmetric orbits. Let the orbit set of an action of SO(n, R) in E % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGOmaaaaaaa!3773!\[\frac{1}{2}\]n(n−1) contain a Plücker submanifold. It is proved that 1) the only symmetric orbits are Plücker orbits and for n = 2ν > 4 the unitary orbits, 2) each of their second order envelopes is trivial, i.e. is a single orbit or its open part.
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References
Boeckx, E.: Foliated semi-symmetric spaces. Thesis. Katholic Univ. Leuven 1994.
Bryant, B.L.; Chern, S.S.; Gardner, R.B.; Goldschmidt, H.L.; Griffiths, P.A.: Exterior Differential Systems. Springer-Verlag, Berlin 1991.
Deprez, J.: Semi-parallel surfaces in Euclidean space. J. Geom. 25 (1985), 192–200.
Dillen, F.; Nölker, S.: Semi-parallelity, multi-rotation surfaces and the helix-property. J. Reine Angew. Math. 435 (1993), 33–63.
Ferus, D.: Symmetric submanifolds of Euclidean space. Math. Ann. 247 (1980), 81–93.
Leichtweiss, K.: Zur Riemannschen Geometrie in Grassmannschen Mannigfaltigkeiten. Math. Z. 76 (1961), 334–366.
Lumiste, Ü.: Semi-symmetric submanifolds with maximal first normal space. Proc. Estonian Acad. Sci., Phys. Math. 38 (1989), 453–457.
Lumiste, Ü.: Semi-symmetric submanifold as the second order envelope of symmetric submanifolds. Proc. Estonian Acad. Sci., Phys. Math. 39 (1990), 1–8.
Lumiste, Ü.: Semi-symmetric submanifolds. Itogi Nauki Tekh., Ser. Probl. Geom. 23 (1991), 1–28 (Russian).
Lumiste, Ü.: Semi-symmetric envelopes of some symmetric cylindrical submanifolds. Proc. Estonian Acad. Sci., Phys. Math. 40 (1991), 245–257.
Lumiste, Ü.: Symmetric orbits of the orthogonal Segre action and their second order envelopes. Rend. Semin. Mat. Messina, Ser. II, 14 (1991) 1, 141–150.
Lumiste, Ü.: Second order envelopes of m-dimensional Veronese submanifolds. Tartu Ülik. Toim., Acta Comm. Univ. Tartu 930 (1991), 35–46.
Lumiste, Ü.: Semi-symmetric submanifolds and modified Nomizu problem. In: N.K. Stephanides (Ed.): Proc. 3rd Congr. of Geom., Thessaloniki (May 26 — June 1, 1991). Aristotle University of Thessaloniki 1992, 263–274.
Lumiste, Ü.: Modified Nomizu problem for semi-parallel submanifolds. In: F. Dillen et al. (Eds.): Geometry and Topology of Submanifolds, VII. Differ. Geom. in honour of Prof. Katsumi Nomizu. World Sc., Singapore [et al.] 1995, 176–181.
Lumiste, Ü.: Symmetric orbits of orthogonal Veronese actions and their second order envelopes. Result. Math. 27 (1995), 284–301.
Nomizu, K.: On hypersurfaces satisfying a certain condition on the curvature tensor. Tôhoku Math. J. 20 (1968), 46–59.
Ros, A.: A characterization of seven compact Kaehler submanifolds by holomorphic pinching. Ann. Math. 121 (1985), 377–382.
Schouten, J. A.: Der Ricci-Kalkül. Springer-Verlag, Berlin 1924.
Shirokov, P.A.: Tensor Calculus. 2nd ed., Izd. Kazan Univ., Kazan 1961 (Russian).
Sinyukov, N.S.: Geodesic maps of Riemannian spaces. Nauka, Moscow 1979 (Russian).
Sternberg, S.: Lectures on differential geometry. Prentice Hall, Inc., Englewood Cliffs, N. J. 1964.
Szabó, Z.I.: Structure theorems on Riemannian spaces satisfying R(X,Y)·R=0,1. J. Differ. Geom. 17 (1982), 531–582.
Takeuchi, M.: Parallel submanifolds of space forms. In: Manifolds and Lie groups. Papers in honor of Yozô Matsushima. Basel 1981, 429–447.
Vilms, J.: Submanifolds of Euclidean space with parallel second fundamental form. Proc. Am. Math. Soc. 32 (1972), 263–267.
Wolf, J.A.: Spaces of constant curvature. Univ. of California, Berkley 1972.
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Lumiste, Ü. Symmetric orbits of orthogonal Plücker actions and triviality of their second order envelopes. Ann Glob Anal Geom 14, 237–256 (1996). https://doi.org/10.1007/BF00054472
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DOI: https://doi.org/10.1007/BF00054472