Renato Tribuzy

A theorem of Hopf and the Cauchy-Riemann inequality

In 1951, Hopf [9] published a theorem in a seminal paper on surfaces of constant mean curvature which can be stated as follows. Let a genus zero compact surface M be immersed in \({\mathbb{R}^3}\) with constant mean curvature H. Then M is isometric to the standard sphere. Hopf gave two proofs of this result (see [9] for details).

Kähler submanifolds with parallel pluri-mean curvature

Abstract We investigate the local geometry of a class of Kahler submanifolds M⊂ R n which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the (1,1)-part (i.e., the dz i d z j -components) of the second fundamental form α, which we call the pluri-mean curvature. We show that these Kahler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic.

Lines of curvature on surfaces immersed in ℝ4

The differential equation of thelines of curvature for immersions of surfaces into ℝ4 is established. It is shown that, for a class of generic immersions of a surface into ℝ4 in theCr-topology,r≥4, all of the umbilic points are locally topologically stable. This type of umbilic points is described.

Lawson correspondence and Ribaucour transformations

We prove that Darboux transformations commute with the Lawson correspondence and we show that the property of completeness is preserved by this commutativity. We provide examples of these results. Two applications provide families of explicitly parametrized complete surfaces of constant mean curvature 1 and − √ 5/2 in H3, depending on 2 parameters and 1 parameter respectively. For special choices of the parameters, we get surfaces that are periodic in one variable and in particular complete cmc surfaces or cmc1 surfaces in H3, with any finite or infinite number of bubbles, “segments” or embedded ends of horosphere type. Moreover, we consider Ribaucour transformations for associated linear Weingarten surfaces in space forms. We show that such a transformation is a Darboux transformation (i.e., it is conformal) if and only if the surfaces have the same constant mean curvature. We prove that Ribaucour transformations for surfaces with constant mean curvature 1 (cmc1) immersed in the hyperbolic space H3 produce embedded ends of horosphere type.

Compatibility of Gauß maps with metrics

Abstract We give necessary and sufficient conditions on a smooth local map of a Riemannian manifold M m into the sphere S m to be the Gaus map of an isometric immersion u : M m → R n , n = m + 1 . We briefly discuss the case of general n as well.

Reduction of the codimension of isometric immersions in space forms

WECONSIDER C” immersionsf: M” --f Q: of an n-dimensional connected manifold M, into an N-dimensional simply connected complete space form Q:, N > n, of constant curvature c. The codimerrsion of the immersion can be reduced to r, if there exists a totally geodesic, (n + r)dimensional submanifold L of Qy such thatf( M) c L. Let xi, . . . , x, be local coordinates in M. The space generated by the derivatives off of al! orders up to k, at aipoint p E M, is the k-th order oscularing space o2ffat p and it is denoted by Osc,. In particular, Osc, is the tangent space T, .LI, of A4 at p, and Osc, is the direct sum of TP M and the subspace generated by the vectors Z(X, 13, x, y E T, M, where CL is the second fundamental form of the immersion. Higher order osculating spaces were introduced by E. Cartan [2] and studied in [l], [6], [7], [9] and [13]. We want to establish sufficient conditions, on the osculating spaces, for reducing the codimension of an immersion. The simplest result in this direction is the classical property of curves in Euclidean space. Namely, if p is a regular curve in R,‘, whose curvatures k I,..., kj_ I do not vanish and kj is identically zero, then /I is contained in an affine j-dimensional subspace of R”. Equivalently, if at every point p of the curve dimOsc, = i, for each i, 1 I i Sj1 and dim O&c, = j1, then we can reduce the codimension to j2. Our main theorems are generalizations of results contained in [14]. Our first theorem shows that for an immersion of a compact manifold A4 “, n 2 2, if the dimension of the k-th order osculating space, k 2 2, is a constant less than n+ k, then we can reduce the codimension.

Parallel mean curvature surfaces in symmetric spaces

We present a reduction-of-codimension theorem for surfaces with parallel mean curvature in symmetric spaces.

A differential equation for lines of curvature on surfaces immersed in ℝ4

We establish the differential equation of the lines of curvature for immersions of surfaces into ℝ4. From the point of view of principal lines of curvature, we show that the differnetila equations under consideration carry almost complete information of the immersed surface.