Abstract.We show that for Wintgen ideal submanifolds in real space forms the (intrinsic) Ricci principal directions and the (extrinsic) Casorati principal directions coincide.
Wintgen Ideal Submanifolds of Real Space FormsLet M n be an n-dimensional Riemannian submanifold of an (n + m)-dimensional real space formM n+m (c) of constant sectional curvature c and let , ∇ and˜ ,∇ be the Riemannian metric and the corresponding LeviCivita connection on M n and onM n+m (c), respectively. Tangent vector fields on M n will be written as X, Y, . . . and normal vector fields on M n inM n+m (c) will be written as ξ, η, . . . . The formulae of Gauss and Weingarten, concerning the decomposition of the vector fields∇ X Y and∇ X ξ, respectively, into their tangential and normal components along M n inM n+m (c), are given by∇ξ, respectively, whereby h is the second fundamental form and A ξ is the shape operator or Weingarten map of M n with respect to the normal vector field ξ, such that˜ (h(X, Y), ξ) = (A ξ (X), Y), and ∇ ⊥ is the connection in the normal bundle.The mean curvature vector field H is defined by H = 1 n tr h and its length H = H is the extrinsic mean curvature of M n inM n+m (c). A submanifold M n inM n+m (c) is totally geodesic when h = 0, totally umbilical when h = H, minimal when H = 0 and pseudo-umbilical when H is an umbilical normal direction. Let {E 1 , . . . , E n , ξ 1 , . . . , ξ m } be any adapted orthonormal local frame field on the submanifold M n inM n+m (c), denoted for short also as {E i , ξ α }, whereby i, j, · · · ∈ {1, 2, . . . , n} and α, β, · · · ∈ {1, 2, . . . , m}.By the equation of Gauss, the (0, 4) Riemann-Christoffel curvature tensor of a submanifold M n inM n+m (c) isand the metrically corresponding (1, 1) tensor or Ricci operator will also be denoted by S: (S(X), Y) = S(X, Y). Since S is symmetric Email addresses: simona.decu@gmail.com (Simona Decu), mirapt@kg.ac.rs (Miroslava Petrović-Torgašev), aleksandar.sebekovic@ftn.kg.ac.rs (AleksandarŠebeković), leopold.verstraelen@wis.kuleuven.be (Leopold Verstraelen)
