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Register a new userConformally flat submanifolds in spheres and integrable systems
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E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S^4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n-2)-spheres with flat normal bundle and constant multiplicities.
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We study conformally flat hypersurfaces $fcolon M^{3} to Q^{4}(c)$ with three distinct principal curvatures and constant mean curvature $H$ in a space form with constant sectional curvature $c$. First we extend a theorem due to Defever when $c=0$ and show that there is no such hypersurface if $H eq 0$. Our main results are for the minimal case $H=0$. If $c eq 0$, we prove that if $fcolon M^{3} to Q^{4}(c)$ is a minimal conformally flat hypersurface with three distinct principal curvatures then $f(M^3)$ is an open subset of a generalized cone over a Clifford torus in an umbilical hypersurface $Q^{3}(tilde c)subset Q^4(c)$, $tilde c>0$, with $tilde cgeq c$ if $c>0$. For $c=0$, we show that, besides the cone over the Clifford torus in $Sf^3subset R^4$, there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions $fcolon M^3 to R^4$ with three distinct principal curvatures of simply-connected conformally flat Riemannian manifolds.
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.
Based on the classical Plucker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in $CP^3$. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues classical theorem of projective geometry. A remarkable characterisation in terms of correlations of $CP^3$ is also recorded.
It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson-Walker warped product, if the gradient of the potential function is non null, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
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