In this paper we study the following problem: To what extent does the type of the Gauss map of a submanifold of W 1 determine the submanifold? Several results in this respect are obtained.In particular, submanifolds with 2-type Gauss map are characterized.Surfaces with ,2-type Gauss map and minimal surfaces of S with 2-type Gauss map are completely classified. Some applications are also given. Introduction.A compact submanifold M of a Euclidean m-space ET is said to be of finite type if the immersion x of M in E" 1 can be expressed as a finite sum of ^-valued eigenfuctions of the Laplacian A of M ,acting on E -valued functions. Minimal submanifolds of a hypersphereReceived 12 March 1986. This work was done while the first author was a C.N.R. visiting professor at the University of Rome. He would like to take this opportunity to express his thanks to the Consiglio Nazionale delle Ricerche of Italy for the invitation and financial support, and also to express his many thanks to his colleagues" at Rome for their hospitality. G(n,m) is canonically imbedded in hu=K,N=(),the notion of finite-type Gauss map is naturally defined.The main purpose of this paper is to study the following problem: To what extent does the type of the Gauss map of a submanifold of ET determine the submanifold?For closed curves in E , the type of a curve in E coincides with that of its Gauss map (Proposition 3.1). In contrast, for submanifolds of dimension £ 2 , the two notions are different.A well-known result of Takahashi says that a compact submanifold of E is of 2-type if and only if it is a minimal submanifold of a hypersphere. In Section 4 we study the following problem: Which submanifolds of E have i-type Gauss map? In this respect, we obtain a chacterization theorem for submanifolds with i-type Gauss map. This result is then applied to obtain some classification theorems of such submanifolds.In Section 5, we show that a standard isometric immersion of an ordinary 2-sphere has 2-type Gauss map if and only if it is not the first standard imbedding. The complete classification of flat minimal tori in with 2-type Gauss map is given in Section 6. In the last section, wegive the complete classification of minimal surfaces of S Preliminaries.Let The following result is known (see [5,7] Gauss Map.Let V be an oriented w-plane in E" 1 . Denote by e ,..

Abstract:The Hermitian symmetric space M = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl 0 (E) → End(TM) mapping Λ 2 E into skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle of End(TM). From this we construct a canonical differential 8-form on EIII, associated with its holonomy Spin(10) · U(1) ⊂ U (16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V (4) ⊂ CP 26 known as the fourth Severi variety.

Abstract. We characterize compact locally conformal parallel G 2 (respectively, Spin (7)) manifolds as fiber bundles over S 1 with compact nearly Kähler (respectively, compact nearly parallel G 2 ) fiber. A more specific characterization is provided when the local parallel structures are flat.

Abstract. We consider compact locally conformal quaternion Kähler manifolds M . This structure defines on M a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project M over a quaternion Kähler orbifold N with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers' structure and obtain restrictions on the Betti numbers. In the homogeneous case N is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of H n −{0} by the diagonal action of appropriately chosen discrete subgroups of CO + (4).

Abstract. For a Spin(9)-structure on a Riemannian manifold M 16 we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form Φ Spin(9) . We then prove that Φ Spin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Φ Spin(9) and Φ 2 Spin(9) in the special case of holonomy Spin(9).

We give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R-16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionic structures on R-2, R-4 and R-8. respectively. (C) 2012 Elsevier Inc. All rights reserved

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Abstract. We consider locally conformal Kähler geometry as an equivariant (homothetic) Kähler geometry: a locally conformal Kähler manifold is, up to equivalence, a pair (K, Γ) where K is a Kähler manifold and Γ a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kähler manifold (K, Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kähler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover we define locally conformal hyperkähler reduction as an equivariant version of hyperkähler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally we show that locally conformal hyperkähler reduction induces hyperkähler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperkähler reduction.