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Affine biharmonic submanifolds in 3-dimensional pseudo-Hermitian geometry. (English) Zbl 1180.58010

Given a Riemannian manifold \((N,h)\) and an affine manifold \((M,\nabla)\), where \(\nabla\) is a linear connection, a map \(\phi:N\to M\) is said to be affine harmonic if the tension field vanish, that is \(\tau(\phi)=\mathrm{trace} \nabla d\phi=0\), generalizing the concept of harmonic map in case \(M\) is Riemannian and \(\nabla\) is the Levi-Civita connection. An affine harmonic variation of \(\phi\) is a family \(\phi_t\) of affine harmonic maps. Its vector variation \(V\), at \(t=0\), defines a Jacobi field along \(\phi\), and satisfies a Jacobi field equation \(\mathcal{J}_{\phi}(V)=0\) that depends on the torsion \(T\) and its first derivative. A map \(\phi\) is said to be affine biharmonic if \(\mathcal{J}_{\phi}(\tau(\phi))=0\).
The second fundamental forms \(\alpha\) and \(\hat{\alpha}\) of an immersed surface into a contact strongly pseudo-convex pseudo Hermitian 3-dimensional Riemannian manifold \(M\), for the Levi-Civita connection and for the Tanaka-Webster connection \(\hat{\nabla}\) are related. Surfaces in a Sasakian 3-manifold with vanishing \(\hat{\alpha}\) are classified, namely the authors conclude they are minimal and anti-invariant ( that is they have zero contact angle). In particular, they conclude that \(M\) is pseudo-Hermitean minimal (\(\mathrm{tr}_h\hat{\alpha} =0\)) iff \(M\) is minimal (\(\mathrm{tr}_h{\alpha} =0\)). Hopf cylinders associated to Boothby-Wang fibrations \(\pi:M \to\bar{M}\) are classified, under the assumption of minimality or of non-minimal biharmonicity (for the case \(M\) is a Sasakian space form). In the later case, if the cylinder is given by \(\Sigma=\pi^{-1}(\bar{\gamma})\), where \(\bar{\gamma}\) is a regular curve, then the curvature of the horizontal lift curve is constant and satisfies \(\kappa^2=H+3\), where \(H\) is the (constant) \(\varphi\)-holomorphic sectional curvature. This leads to a non-existence theorem for proper pseudo-Hermitean biharmonic Hopf cylinders when \(H\leq -3\), and the conclusion of uniqueness of such cylinder as the Clifford tori of constant mean curvature \(\pm 1\) in \(\mathbb{S}^3\).

MSC:

58E20 Harmonic maps, etc.
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