It is well known that totally geodesic Lagrangian submanifolds of a complexspace-form f M n (4c) of constant holomorphic sectional curvature 4c are real-spaceforms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form f M n (4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1 I1 fk Ik 1 N n k (c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simplyconnected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-spaceform f M n (4c). Conversely, if L: M n (c) ! f M n (4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-formM n (c) of constant sectional curvaturec into a complex-space-form f M n (4c), thenM n (c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

Lagrangian isometric immersions of a real-space-form M n ( c ) into a complex-space-form M ˜ n (4 c )

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 to have pointwise 1-type Gauss map.

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Tensor product surfaces with pointwise 1-type gauss map

The structure of pointwise slant submanifolds in an almost product Riemannian manifold is investigated and the special proper pointwise slant surfaces of a locally product manifold are introduced. A relation involving the squared mean curvature and the Gauss curvature of pointwise slant surface of a locally product manifold is proved. Two examples of proper pointwise slant surfaces of a locally product manifold, one of which is special and the other one is not special, are given.

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Special proper pointwise slant surfaces of a locally product Riemannian manifold

In this article we investigate Vranceanu rotation surfaces with pointwise 1- type Gauss map in Euclidean 4-space \( \mathbb{E}^4 \). We show that a Vranceanu rotation surface M has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficent conditions for Vranceanu rotation surface to have pointwise 1-type Gauss map.

Vranceanu surface in \( \mathbb{E}^4 \) with pointwise 1- type Gauss map

For a totally real surface M of a complex space form f M(4c) of arbitrary codimen- sion, we obtain an inequality relating the squared mean curvature kHk 2 , the holomorphic sectional curvature c, the Gauss curvature K and the elliptic curvature K E of the surface. Using the notion of ellipse of curvature, we obtain a characterization of the equality. An example of a Lagrangian surface of C 2 satisfying the equality case is given.

An inequality for totally real surfaces in complex space forms

Recently B.Y. CHEN initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold [3]. In [6] the particular case of tensor product of two Euclidean plane curves was studied. The minimal one were classified, and necessary and sufficient conditions for such a tensor product to be totally real or complex or slant were established. In the present paper we study for tensor product of Euclidean plane curves the problem of B.Y. CHEN: to what extent do the properties of the tensor product immersion f ⊗ h of two immersions f, h determines the immersions f, h ? [3]

Bernard Rouxel

Papers

Tensor product surfaces of euclidean plane curves