We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkähler and G2-man...

In this paper we investigate the geometry of Calibrated submanifolds and study relations between their moduli-space and geometry of the ambient manifold. In particular for a Calabi-Yau manifold we define Special Lagrangian submanifolds for any Kahler metric on it. We show that for a choice of Kahler metric the Borcea-Voisin threefold has a fibration structure with generic fiber being a Special ...

The study of submanifolds with harmonic mean curvature vector field was initiated by B. Y. Chen in 1985 and arose in the context of his theory of submanifolds of finite type. For a survey on submanifolds of finite type and various related topics was presented in [8, 9]. Let M r be an n-dimensional, connected submanifold of the pseudo-Euclidean space E s . Denote by x⃗, H⃗, and △ respectively the ...

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit “hidden symmetry” from conformal KillingYano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimensiontwo submanifold with constant normalized null expansion (null mean curva...

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit “hidden symmetry” from conformal KillingYano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimensiontwo submanifold with constant normalized null expansion (null mean curva...

in this paper we consider contact cr-warped product submanifolds of the type $m = n_ttimes_f n_perp$, of a nearly kenmotsu generalized sasakian space form $bar m(f_1‎, ‎f_2‎, ‎f_3)$ and by use of hopf's lemma we show that $m$ is simply contact cr-product under certain condition‎. ‎finally‎, ‎we establish a sharp inequality for squared norm of the second fundamental form and equality case i...

Infinitesimal deformations and stabilities of singular Legendre submanifolds.

In literature (Kobayashi and Nomizu, 1963, 1969; Yano and Ako, 1972; Ishihara, 1974; Özdemir, 2006; Alagöz et al., 2012), almost complex and almost quaternionic structures have been investigated widely. These structures are special structures on the tangent bundle of a manifold. A detailed review can be found in Kirichenko and Arseneva (1997). Let us recall some basic facts and definitions from...

In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs C∞ deformations of coisotropic submanifolds and define the corresponding C∞-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation sp...