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Let $fcolon M^{2n}tomathbb{R}^{2n+ell}$, $n geq 5$, denote a conformal immersion into Euclidean space with codimension $ell$ of a Kaehler manifold of complex dimension $n$ and free of flat points. For codimensions $ell=1,2$ we show that such a submanifold can always be locally obtained in a rather simple way, namely, from an isometric immersion of the Kaehler manifold $M^{2n}$ into either $mathbb{R}^{2n+1}$ or $mathbb{R}^{2n+2}$, the latter being a class of submanifolds already extensively studied.
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Let $fcolon M^{2n}tomathbb{R}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $ngeq 2$ into Euclidean space with codimension $p$. If $2pleq 2n-1$, we show that generic rank conditions on the second fundamental form of the submanifold imply that $f$ has to be a minimal submanifold. In fact, for codimension $pleq 11$ we prove that $f$ must be holomorphic with respect to some complex structure in the ambient space.
This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds is a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Euclidean hypersurfaces that admit nontrivial conformal variations. Our first main result is a Fundamental theorem for conformal infinitesimal variations. The second is a rigidity theorem for Euclidean submanifolds that lie in low codimension.
In this paper we investigate $m$-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least $m-2$ at any point. These are austere submanifolds in the sense of Harvey and Lawson cite{harvey} and were initially studied by Bryant cite{br}. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit cite{DF2} in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of $1$-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply-connected ones.
In this paper we give local and global parametric classifications of a class of Einstein submanifolds of Euclidean space. The highlight is for submanifolds of codimension two since in this case our assumptions are only of intrinsic nature.
Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in [17], it is interesting to find their existence. In this paper, we prove the existence of warped product semi-slant submanifolds of nearly Kaehler manifolds by a characterization. To this end we obtain an inequality for the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also discussed.
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