New ideas in differential geometry of submanifolds (Q2783460)

These lecture notes deal with two modern streams of differential geometry: the Grassmann image of submanifolds, and isometric immersions of \(n\)-dimensional Lobachevsky space \(L^n\) into \((2n-l)\)-dimensional Euclidean space \(E^{(2n-l)}\). The author describes the connection between the theory of the isometric immersions of the Lobachevsky space \(L^3\) into \(E^5\) and the classical problem from mechanics about the motion of a rigid body with a fixed point in a central field of gravitation.NEWLINENEWLINEThe metric on the Grassmann manifolds and its sectional curvatures are investigated based on results by \textit{K. Leichtweiss} [Math. Z. 76, 334--366 (1961; Zbl 0113.37102)] and \textit{Y.-C. Wong} [Proc. Natl. Acad. Sci. USA 60, 75--79 (1968; Zbl 0169.23904)]. Then the author gives some applications of this theory to the theory of the Grassmann image. A classification of submanifolds in Euclidean spaces with respect to the Grassmann image is given.NEWLINENEWLINEThe table of contents gives some impression of the book:NEWLINENEWLINE Various ways to generalization of the notion of a spherical image, Grassmann image of 2-dimensional surfaces \(F^2\) in \(E^4\), Representations of points of the Grassmann manifolds with the help of Plücker coordinates and matrices, The metric of the Grassmann image, Some properties of the Grassmann mapping of immersion of the Lobachevsky space \(L^n\) in \(E^{(2n-1)}\), Angles between two \(k\)-dimensional spaces, Geodesic curves of the Grassmann manifold, The Grassmann image of the 3-dimensional analogy of the pseudosphere, K. Leichtweiss and Y.-C. Wong. The formula for the curvature of Grassmann manifolds. Tangent vectors of the Grassmann image. The sectional curvature of \(G_{k,n+k}\) for a square tangent to the Grassmann image. The curvature \(\overline{K}\) for submanifolds with flat normal connection. The curvature \(\overline{K}(\sigma)\) for 2-dimensional surface \(F^2\subset F^4\). The expression of \(\overline{K}\) in terms of the curvatures of the tangent and normal connections. Once again about the metric of the Grassmann manifold, On the local immersions of the Lobachevsky space \(L^n\) in \(E^{(2n-1)}\). Initial data and local analytical solutions of the `LE' system, Classification of submanifolds in the Euclidean spaces with respect to the Grassmann image, Connection between the form of the local projection of 2-dimensional surface in \(E^4\) and the type of its Grassmann image. Another classification and the notion of \(k\)-saddle submanifolds. Model of a gauge field. Indicatrix of normal curvature, The Grassmann image of the submanifold \(L^3\subset L^5\). Immersions with hyperplane Grassmann image. New form for the basic system. Isometric immersions of \(L^3\) in \(E^5\) and the motion of a solid body with fixed point. Little poem about first integral. Higher dimensional generalizations of the Bianchi and Bäcklund transformations.