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.
We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $fcolon M^n_ctoQ^{n+p}_{tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature $tilde c$, and free
of weak-umbilic points if $c>tilde{c}$. We show that the substantial codimension of $f$ is $p=n-1$ if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank $n-1$. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit B{a}cklund and Ribaucour transformations.
We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.
In the realm of conformal geometry, we give a classification of the Euclidean hypersurfaces that admit a non-trivial conformal infinitesimal variation. In the restricted case of conformal variations, such a classification was obtained by E. Cartan in
1917. The case of infinitesimal isometric variations was done by U. Sbrana in 1908. In particular, we show that the class of hypersurfaces that allow a conformal infinitesimal variation is much larger than the one considered by Cartan.
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 geometr
y. 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.
This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of isometric imme
rsions in Euclidean space, we prove that a system of three equations for a certain pair of tensors are the integrability conditions for the differential equation that determines the infinitesimal variations. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds.
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 subman
ifold 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.
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.
A basic question in submanifold theory is whether a given isometric immersion $fcolon M^ntoR^{n+p}$ of a Riemannian manifold of dimension $ngeq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine infinitesimal bend
ing. That is, if there exists a genuine smooth variation of $f$ by immersions that are isometric up to the first order. Until now only the hypersurface case $p=1$ was well understood. We show that a strong necessary local condition to admit such a bending is the submanifold to be ruled and give a lower bound for the dimension of the rulings. In the global case, we describe the situation of compact submanifolds of dimension $ngeq 5$ in codimension $p=2$.
Austere submanifolds of Euclidean space were introduced in 1982 by Harvey and Lawson in their foundational work on calibrated geometries. In general, the austerity condition is much stronger than minimality since it express that the nonzero eigenvalu
es of the shape operator of the submanifold appear in opposite pairs for any normal vector at any point. Thereafter, the challenging task of finding non-trivial explicit examples, other than minimal immersions of Kaehler manifolds, only turned out submanifolds of rank two, and these are of limited interest in the sense that in this special situation austerity is equivalent to minimality. In this paper, we present the first explicitly given family of austere non-Kaehler submanifolds of higher rank, and these are produced from holomorphic data by means of a Weierstrass type parametrization.
