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Register a new userConformal infinitesimal variations of submanifolds
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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.
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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 immersions 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.
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.
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 bending. 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 $p=2$.
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.
In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $mathbb{C}[partial]$-modules such that both of them are subalgebras and the natural conformal bilinear form is invariant. The coboundary case leads to the introduction of associative conformal Yang-Baxter equation whose antisymmetric solutions give antisymmetric infinitesimal conformal bialgebras. Moreover, the construction of antisymmetric solutions of associative conformal Yang-Baxter equation is given from $mathcal{O}$-operators of associative conformal algebras as well as dendriform conformal algebras.
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