Some characterizing results for hemi-slant warped product submanifolds of a Kaehler manifold

Many differential geometric properties of submanifolds of a Kaehler manifold are looked into via canonical structure tensors P and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product (i.e. locally a Riemannian product of a holomorphic and a totally real submanifold) if and only if the canonical tensor P is parallel on the submanifold. Since, warped product manifolds are generalized version of Riemannian product of manifolds, in this article, we consider the covariant derivatives of the structure tensors on a hemi-slant submanifold of a Kaehler manifold. Our investigations have led us to characterize hemi-slant warped product submanifolds.

The Hele-Shaw flow and moduli of holomorphic discs

We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

TOTAL REALITY OF CONORMAL BUNDLES OF HYPERSURFACES IN ALMOST COMPLEX MANIFOLDS

A generalization to the almost complex setting of a well-known result by Webster is given. Namely, we prove that if Γ is a strongly pseudoconvex hypersurface in an almost complex manifold (M, J), then the conormal bundle of Γ is a totally real submanifold of (T* M, 𝕁), where 𝕁 is the lifted almost complex structure on T* M defined by Ishihara and Yano.

Totally real pseudo-umbilical submanifolds of a quaternion space form

Let M(c) denote a 4n-dimensional quaternion space form of quaternion sectional curvature c, and let P(H) denote the 4n-dimensional quaternion projective space of constant quaternion sectional curvature 4. Let N be an n-dimensional Riemannian manifold isometrically immersed in M(c). We call N a totally real submanifold of M(c) if each tangent 2-plane of N is mapped into a totally real plane in M (c). B. Y. Chen and C. S. Houh proved in [1].

On Totally Real Submanifolds in a 6-Sphere

The 6-dimensional sphere S6 has an almost complex structure induced by properties of Cayley algebra. With respect to this structure S6 is a nearly Kaehlerian manifold. We investigate 2-dimensional totally real submanifolds in S6. We prove that a 2-dimensional totally real submanifold in S6 is flat.

Chen's problem on mixed foliate CR-submanifolds

We prove that the simply connected compact mixed foliate CR-submanifold in a hyperbolic complex space form is either a complex submanifold or a totally real submanifold. This is the problem posed by Chen.

Totally umbilicalCR-submanifolds of Semi-Riemannian Kaehler manifolds

We study totally umbilicalCR-submanifolds of a Kaehler manifold carrying a semi-Riemannian metric. It is shown that for dimension of the totally real distribution greater than one, these submanifolds are locally decomposable into a complex and a totally real submanifold of the Kaehler manifold. For dimension equal to one, we show, in particular, that they are endowed with a normal contact metric structure if and only if the second fundamental form is parallel.