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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.
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We address the problem of determining the hypersurfaces $fcolon M^{n} to mathbb{Q}_s^{n+1}(c)$ with dimension $ngeq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $sin {0, 1}$ for which there exists another isometric immersion $tilde{f}colon M^{n} to mathbb{Q}^{n+1}_{tilde s}(tilde{c})$ with $tilde{c} eq c$. For $ngeq 4$, we provide a complete solution by extending results for $s=0=tilde s$ by do Carmo and Dajczer and by Dajczer and the second author. Our main results are for the most interesting case $n=3$, and these are new even in the Riemannian case $s=0=tilde s$. In particular, we characterize the solutions that have dimension $n=3$ and three distinct principal curvatures. We show that these are closely related to conformally flat hypersurfaces of $mathbb{Q}_s^{4}(c)$ with three distinct principal curvatures, and we obtain a similar characterization of the latter that improves a theorem by Hertrich-Jeromin. We also derive a Ribaucour transformation for both classes of hypersurfaces, which gives a process to produce a family of new elements of those classes, starting from a given one, in terms of solutions of a linear system of PDEs. This enables us to construct explicit examples of three-dimensional solutions of the problem, as well as new explicit examples of three-dimensional conformally flat hypersurfaces that have three distinct principal curvatures.
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.
The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless $kappa$-Einstein soliton and is realized on the cotangent bundle of an affine surface.
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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