By Ülo Lumiste

Quite easily, this e-book deals the main complete survey thus far of the idea of semiparallel submanifolds. It starts off with the required historical past fabric, detailing symmetric and semisymmetric Riemannian manifolds, tender manifolds in area kinds, and parallel submanifolds. The ebook then introduces semiparallel submanifolds and provides a few characterizations for his or her category in addition to numerous subclasses. The insurance strikes directly to talk about the concept that of major symmetric orbit and provides all recognized effects pertaining to umbilic-like major symmetric orbits. With greater than forty released papers less than his belt at the topic, Lumiste offers readers with the main authoritative treatment.

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**Extra info for Semiparallel Submanifolds in Space Forms: Envelopes of Symmetric Orbits **

**Example text**

Let this point be identified, as above, with its radius 3 For the classical case, when c = 0, m = 2, and n = 3, this identity was first given in a preliminary form in 1853 by K. Peterson in his Dorpat (now Tartu, Estonia) dissertation, then in 1856 by G. Mainardi, and in a more modern form in 1860 by D. Codazzi (see [GLOP 70], [Rei 73], [Ph 79], [Lu 97a]). , [Ka 48], Sections 52 and 55, where this tensor in the classical case is denoted by πij k . 30 2 Submanifolds in Space Forms vector from a fixed origin o in σ E n+1 (if c = 0 this o is the center of the standard model of the space or spacetime form).

Ca 60]; its curvature 2-forms are considered as components of the torsion of a submanifold) and investigated afterwards by D. I. Perepelkin [Per 35], F. FabriciusBierre [Fa 36], among others. For submanifolds L. van der Waerden [vdWa 27] and E. Bortolotti [Bo 27] worked out a special notation scheme, called the D-symbolics in [SchStr 35]. Subsequently, ¯ and called the van der Waerden–Bortolotti the pair of ∇ and ∇ ⊥ was denoted by ∇ connection of the submanifold M m in N n (c) (see [Ch 73b], [Lu 2000a]).

Is } and contains dx + 2 d x + · · · s! d x. In general x both of these subspaces are (pseudo-)Euclidean. The orthogonal complement of (s−1) O ∗ M m in (s) O ∗ M m is called the sth-order outer normal subspace (s) T ∗⊥ M m x x x of M m at x. , [CGR 90] for the Euclidean case E n = 0 N n (0)). A submanifold M m in s N n (c) is said to be regular if its normal subspaces of all orders have constant dimensions, denoted below by m1 , . . , ms , . . , and are (pseudo-)Euclidean. Then the frame bundle adapted to M m can be specialized by means of these subspaces as follows.