In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalr flat metric having constant mean curvature. This result can be seen as a generalization to higher dimensions of the well known Riemann mapping Theorem in the plane.

The geometric form of a conjecture associated with the names of Loewner and Carath eodory states that near an isolated umbilic in a smooth surface in R 3 , the principal line elds must have index 1. Real solutions of the diierential equation @ 2 z ! = g , where the complex function g is given only up to multiplication by a positive function, are intimately related to umbil-ics. We determine nec...

We first prove some results on invariant lightlike submanifolds of indefinite Sasakian manifolds. Then, we introduce a general notion of contact Cauchy-Riemann (CR) lightlike submanifolds and study the geometry of leaves of their distributions. We also study a class, namely, contact screen Cauchy-Riemann (SCR) lightlike submanifolds which include invariant and screen real subcases. Finally, we ...

Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are obtained by complex submanifolds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: indeed all special Lagrangian submanifolds of X turn out to...

Given a real vector space V equipped with an Euclidean metric, (after rescaling) any p-form φ ∈ V p V ∗ defines a calibration on V . This note identifies an exterior differential system whose integral submanifolds are precisely the critical submanifolds of the calibration. In particular, calibrated submanifolds are necessarily integral submanifolds of the system. The result is extended to calib...

In the present paper we study lightlike submanifolds of almost paracontact metric manifolds. We define invariant lightlike submanifolds. We study radical transversal lightlike submanifolds of para-Sasakian manifolds and investigate the geometry of distributions. Also we introduce a general notion of paracontact Cauchy-Riemann (CR) lightlike submanifolds and we derive some necessary and sufficie...

In this paper we study the pairs of orthogonal foliations on oriented surfaces immersed in R3 whose singularities and leaves are, respectively, the umbilic points and the lines of normal mean curvature of the immersion. Along these lines the immersions bend in R3 according to their normal mean curvature. By analogy with the closely related Principal Curvature Configurations studied in [S-G], [G...

A submanifold of a Euclidean space is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. The notion of constant-ratio submanifolds was first introduced and studied by the author in [5, 8] during the early 2000s. Such submanifolds relate to a problem in physics concerning the motion in a central force field...

Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed curve F in the complex plane, we introduce the notion of the co...

Assuming the ambient manifold is KK ahler, the theory of complex sub-manifolds can be placed in the more general context of calibrated submanifolds, see HL]. It is therefore natural to try to extend some of the many results in complex geometry to the other calibrated geometries of HL]. In particular, the question of deformability of calibrated submanifolds is addressed here (analogous to Kodair...