Symmetric orbits of orthogonal veronese actions and their second order envelopes

The orbits of Lie groups acting in Euclidean spaces by isometries are extrinsically symmetric iff they are parallel, i.e. satisfy ${\overline \nabla} h =0 $ Submanifolds characterized by the integrability condition $ {\overline R}\o\ h =0 $ of this system ${\overline \nabla} h =0$ are called semi-parallel (or semi-symmetric, extrinsically); they are the 2nd order envelopes of the symmetric orbits. For the orthogonal Veronese action, corresponding to the map well-known from the algebraic geometry, all symmetric orbits will be determined, as well as their 2nd order envelopes. The results are essential for the classification of the semi-parallel submanifolds.     

Distributions on homogeneous spaces

A survey of results and ideas in the general analytic treatment of the theory of distributions on homogeneous spaces is presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 5–24, 1977.     

Normal connection and submanifolds with parallel normal fields in a space of constant curvature

A summary account of the cycle of papers of the authors on submanifolds of a space of constant curvature with a parallel field of p-directions, also containing some new results. In the concluding section there is given a survey of papers on submanifolds of a Riemannian manifold using the concept of normal connection.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp. 3–30, 1981.     

Symmetric orbits of orthogonal Plücker actions and triviality of their second order envelopes

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 = 2 > 4 the unitary orbits, 2) each of their second order envelopes is trivial, i.e. is a single orbit or its open part.Partially supported by ESF Grant 139/305     

Differential-geometric structures and Gauge theories

Attention is given basically to the construction of auxiliary bundles and the development of a right-invariant formalism: the Lie algebra of right-invariant vertical vector fields cal P instead of the algebra of fundamental fields fund P, the corresponding 1-forms as means for interpreting Faddeev-Popov ghosts.The content of the plenary report of the author Differential-Geometric Methods in Gauge Theories to the Seventh All-Union Geometry Conference (Odessa, September 18–19, 1984) is recounted in extended and supplemented form.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 17, pp. 153–171, 1985.     

Semisymmetric submanifolds

This article surveys new work on semisymmetric and k-parallel submanifolds M^m in Eⁿ and Mⁿ(c).Translated from Itogi Nauki i Tekhniki Seriya Problemy Geometrii, Vol. 23, pp. 3–28, 1991.     

Absence of complete Lobachevskii planes in two classes of surfaces in E4 with K=−1

Translated from Matematicheskie Zametki, Vol. 48, No. 1, pp. 68–74, July, 1990.     

Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

A Riemannian manifold is said to be semisymmetric if R(X, Y) · R = 0. A submanifold of Euclidean space which satisfies $\bar R\left( {X,Y} \right)$ is called semiparallel. It is known that semiparallel submanifolds are intrinsically semisymmetric. But can every semisymmetric manifold be immersed isometrically as a semiparallel submanifold? This problem has been solved up to now only for the dimension 2, when the answer is affirmative for the positive Gaussian curvature. Among semisymmetric manifolds a special role is played by the foliated ones, which in the dimension 3 are divided by Kowalski into four classes: elliptic, hyperbolic, parabolic and planar. It is shown now that only the planar ones can be immersed isometrically into Euclidean spaces as 3-dimensional semiparallel submanifolds. This result is obtained by a complete classification of such submanifolds.