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We study locally strongly convex, hyperbolic centroaffine Tchebychev hypersurfaces with complete centroaffine metric. We give a classification in case that the centroaffine Ricci tensor is semi-positive and the scalar curvature is constant. The classification extends known global results for hyperbolic affine hyperspheres and for complete centroaffine extremal hypersurfaces.

We give a complete classification of totally umbilical, parallel and semi-parallel hypersurfaces of the Riemannian product space \( \mathbb{S}^n \) × ℝ.

There exists a well known construction which allows to associate with two hyperbolic affine hyperspheres $$f_{i} : M^{n_{i}}_{i} \rightarrow {\mathbb{R}}^{n_{i}+1}$$ a new hyperbolic affine hypersphere immersion of $$I \times M_{1} \times M_{2}$$ into $${\mathbb{R}}^{n_{1}+n_{2}+2}$$ . In this paper we deal with the inverse problem: how to determine from properties of the difference tensor whether a given hyperbolic affine hypersphere immersion of a manifold $$M^{n} \rightarrow R^{n+1}$$ can be decomposed in such a way.

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic., Comment: 15 pages

© 2016 World Scientific Publishing Company. We study affine hypersurfaces M, which have isotropic difference tensor. Note that, any surface always has isotropic difference tensor. In case that the metric is positive definite, such hypersurfaces have been previously studied in [O. Birembaux and M. Djoric, Isotropic affine spheres, Acta Math. Sinica 28(10) 1955-1972.] and [O. Birembaux and L. Vrancken, Isotropic affine hypersurfaces of dimension 5, J. Math. Anal. Appl. 417(2) (2014) 918-962.] We first show that the dimension of an isotropic affine hypersurface is either 5, 8, 14 or 26. Next, we assume that M is an affine hypersphere and we obtain for each of the possible dimensions a complete classification. ispartof: International Journal of Mathematics vol:27 issue:9 pages:1-41 status: published

© 2016 Elsevier B.V. In this paper, we study the Lagrangian submanifolds in the homogeneous nearly Kähler S3×S3 with parallel second fundamental form. We first prove that every Lagrangian submanifold with parallel second fundamental form in any 6-dimensional strict nearly Kähler manifold is totally geodesic. Then we give a complete classification of the totally geodesic Lagrangian submanifolds in the homogeneous nearly Kähler S3×S3. publisher: Elsevier articletitle: Lagrangian submanifolds in the 6-dimensional nearly Kähler manifolds with parallel second fundamental form journaltitle: Journal of Geometry and Physics articlelink: http://dx.doi.org/10.1016/j.geomphys.2016.06.004 content_type: article copyright: © 2016 Elsevier B.V. All rights reserved. ispartof: Journal of Geometry and Physics vol:108 pages:21-37 status: published

There is an almost complex structure J on the sphere \(S^6(1)\) defined by multiplication of the Cayley numbers. This structure is nearly Kahler. A submanifold of a manifold with an almost complex structure is CR, by Bejancu, if it has a differentiable holomorphic distribution \(\mathcal H\) such that its orthogonal complement \(\mathcal H^\perp \subset TM\) is a totally real distribution. A CR-submanifolds of \(S^6(1)\) has to be at least three-dimensional, so with disregarding the hypersurfaces which are trivially CR in the focus of investigation are three and four dimensional submanifolds. We give examples of such submanifolds, show the existence and uniqueness theorem for the three dimensional case, and present the results concerning \(\mathcal H\) and \(\mathcal H^\perp \) totally geodesic submanifolds. We also give examples obtained from the almost contact manifolds. In the four dimensional case, we show the classification of CR minimal submanifolds that satisfy Chen’s basic equality and of those that are not linearly full in \(S^6(1)\).

© 2016, University of Nis. All rights reserved. In this note we are interested in isotropic totally real Lorentzian submanifolds of indefinite complex space forms. We show that such submanifolds are alwaysH-umbilical warped product immersions and we determine also the warping function. ispartof: Filomat vol:30 issue:10 pages:2857-2867 status: published

In this article we study surfaces in \( {\Bbb S}^2\times {\Bbb R} \) for which the unit normal makes a constant angle with the \( {\Bbb R} \)-direction. We give a complete classification for surfaces satisfying this simple geometric condition.

We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches, assuming the dimension is 2 and the surface is definite, a complete classification follows.

In this paper, we classify 3-dimensional minimal CR submanifolds M of the nearly Kahler 6-dimensional sphere which satisfy Chen’s basic equality, i.e. δ M ( p ) = 2 , where δ M ( p ) = τ ( p ) − inf K ( p ) , for every p ∈ M .

In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6–sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.

LetN n (4c) be ann-dimensional complex space form of constant holomorphic sectional curvature 4c and letx:M n →N n (4c) be ann-dimensional Lagrangian submanifold inN n (4c). We prove that the following inequality always hold onM n: $$\left| {\bar \nabla h} \right|^2 \geqslant \frac{{3n^2 }}{{n + 2}}\left| {\nabla ^ \bot \vec H} \right|^2 $$ whereh is the second fundamental form andH is the mean curvature of the submanifold. We classify all submanifolds which at every point realize the equality in the above inequality. As a direct consequence of our Theorem, we give, a new characterization of theWhitney spheres in a complex space form.

We study locally strongly convex surfaces with complete flat Blaschke metric. We show how we can characterize all known examples by a tensorial condition involving the covariant derivative of the shape operator and the gradient of the Pick invariant.

We study affine immersions as introduced by Nomizu and Pinkall. We classify those affine immersions of a surface in 4-space which are degenerate and have vanishing cubic form (i.e. parallel second fundamental form). This completes the classification of parallel surfaces of which the first results were obtained in the beginning of this century by Blaschke and his collaborators., Comment: 16 pages, 1 figure (uses epsf.tex), submitted for publication

On a non-degenerate hypersurface it is well known how to induce an affine connection ∇ and a symmetric bilinear form, called the affine metric. Conversely, given a manifold M and an affine connection ∇ one can ask whether this connection is locally realizable as the induced affine connection on a nondegenerate affine hypersurface and to what extend this immersion is unique. In case that the image of the curvature tensor R of ∇ is 2-dimensional and M is at least 3-dimensional a rigidity theorem was obtained in [4]. In this paper, we discuss positive definite n-dimensional affine hypersurfaces with rank 1 shape operator (which is equivalent with 1-dimensional image of the curvature tensor) which are non-rigid. We show how to construct such affine hypersurfaces using solutions of (n - 1)-dimensional differential equations of Monge–Ampère type.

A surface x: M → Sn is called a Willmore surface if it is a criticalsurface of the Willmore functional ∫M (S − 2H2)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S1(1) and a particularsmall circle in S2(1), and therefore is contained in S5(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in Sn(1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS1(1), whereas the other one is contained either in S2(1) or in S3(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S5. Also in the latter casewe explicitly include examples.

It is well known that the sphere S6(1) admits an almost complex structure J, constructed using the Cayley algebra, which is nearly Kahler. Let M be a Riemannian submanifold of a manifold \({\widetilde{M}}\) with an almost complex structure J. It is called a CR submanifold in the sense of Bejancu (Geometry of CR Submanifolds. D. Reidel Publ. Dordrecht, 1986) if there exists a C∞-differentiable holomorphic distribution \({\mathcal D_1}\) in the tangent bundle such that its orthogonal complement \({\mathcal D_2}\) in the tangent bundle is totally real. If the second fundamental form vanishes on \({\mathcal D_i}\), the submanifold is \({\mathcal D_i}\)-geodesic. The first example of a three-dimensional CR submanifold was constructed by Sekigawa (Tensor N S 41:13–20, 1984). This example was later generalized by Hashimoto and Mashimo (Nagoya Math J 156:171–185, 1999). Note that both the original example as well as its generalizations are \({\mathcal D_1}\)- and \({\mathcal D_2}\)-geodesic. Here, we investigate the class of the three-dimensional minimal CR submanifolds M of the nearly Kahler sphere S6(1) which are not linearly full. We show that this class coincides with the class of \({\mathcal D_1}\)- and \({\mathcal D_2}\)- geodesic CR submanifolds and we obtain a complete classification of such submanifolds. In particular, we show that apart from one special example, the examples of Hashimoto and Mashimo are the only \({\mathcal D_1}\)- and \({\mathcal D_2}\)-geodesic CR submanifolds.

In this paper we initiate the study of almost complex surfaces in the nearly Kahler $S^3\times S^3$. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kahler $S^3\times S^3$. We also find a local correspondence between almost complex surfaces in the nearly Kahler $S^3\times S^3$ and solutions of the general $H$-system equation introduced by Wente ([13]), thus obtaining a geometric interpretation of solutions of the general $H$-system equation. From this we deduce a correspondence between constant mean curvature surfaces in $\mathbb R^3$ and almost complex surfaces in the nearly Kahler $S^3\times S^3$ with vanishing holomorphic differential. This correspondence allows us to obtain a classification of the totally geodesic almost complex surfaces. Moreover, we prove that almost complex topological 2-spheres in $S^3\times S^3$ are totally geodesic. Finally, we also show that every almost complex surface with parallel second fundamental form is totally geodesic.

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

It is well-known that locally strongly convex affine hyperspheres can be determinedas solutions of differential equations of Monge-Ampere type. In this paper we study in particular the 3-dimensional case and we assume that the hypersphere admits a Killing vector field (with respect to the affine metric) whose integral curves are geodesics with respect to both the induced affine connection and the Levi-Civita connection of the affine metric. We show that besides the already known examples, such hyperspheres can be constructed starting from the 2-dimensional Poisson equation, the 2-dimensional sine-Gordon equation or the 2-dimensional cosh-Gordon equation.

It is well-known that the only minimal Lagrangian submanifolds of constant sectional curvature $c$ in a Riemannian complex space form of constant holomorphic sectional curvature $4c$ are the totally geodesic ones. In this paper we investigate minimal Lagrangian Lorentzian submanifolds of constant sectional curvature $c$ in Lorentzian complex space form of constant holomorphic sectional curvature $4c$. We prove that the situation in the Lorentzian case is quite different from the Riemannian case. Several existence and classification theorems in this respect are obtained. Some explicit expression of flat minimal Lagrangian submanifolds in flat complex Lorentzian space form are also presented.

We study affine immersions, as introduced by Nomizu and Pinkall, of $M^n$ into $\R^$. We call $M^n$ linearly full if the image of $M$ is not contained in a lower dimensional affine space. Typical examples of affine immersions are the Euclidean and semi-Riemannian immersions. A classification, under an additional assumption that the rank of the second fundamental form is at least two, of the hypersurfaces with parallel second fundamental form was obtained by Nomizu and Pinkall. If we assume that the second fundamental form is parallel and $M$ is linearly full, then $p \le n(n+1)/2$. In this paper we completely classify the affine immersions with parallel second fundamental form in $\R^$, obtaining amongst others the generalized Veronese immersions.

We study 3-dimensional minimal Lagrangian submanifolds of the 3-dimensional complex projective space ℂP3 (4) which admit a unit length Killing vector field whose integral curves are geodesics. We show that such Lagrangian submanifolds can be obtained from either horizontal holomorphic curves in ℝP3 (4) (or equivalently superminimal immersions of surfaces in S4 (1)) or from solutions of the two dimensional sinh-Gordon equation. In the latter case, we explicitly obtain the immersions in terms of elliptic functions in the case that the solutions of the sinh-Gordon equation depend only on one variable.

We completely classify affine surfaces with projectively flat induced connection and flat affine metric.

There are some obscurity and mistake in [2]. In this note we clarify the obscurity by providing a detailed proof and make a correction on the mistake.

In this paper we characterize affine translation surfaces with constant Gaussian curvature. We show that such surfaces must be flat and that one of the defining curves must be planar.

Considering odd-dimensional complex projective space as a complex contact manifold, one may ask which of the Calabi (Veronese) imbeddings can be positioned by a holomorphic congruence as integral submanifolds of the complex contact structure. It is first shown that when the first normal space is the whole normal space, this is impossible. It is also shown to be impossibile for a Calabi surface (complex dimension 2) in complex projective space of dimension 9 where one has both a first and second normal space. However when the complex dimension of the submanifold is odd and the whole normal space consists of the first and second normal spaces, then there is a holomorphic congruence positioning the Calabi imbedding as an integral submanifold of the complex contact structure.

We study surfaces M 2 in the four-dimensional affine space $$R^4 $$ equipped with its usual torsion-free connection D and parallel volume form given by the determinant.

In this paper we study nondegenerate affine surfaces in the 4-dimensional affine space $$\mathbb{R}^4 $$ . We assume that both the connection ∇ and the normal connection ∇⊥ induced by the canonical equiaffine transversal bundle are flat. Surfaces with constant equiaffine transversal bundle are trivial examples of such surfaces. Here, we obtain a complete classification of all such surfaces which do not have constant equiaffine normal bundle.

In this paper we prove the Magid-Ryan conjecture for 4-dimensional affine hyperspheres in R5. This conjecture states that every affine hypersphere with non-zero Pick invariant and constant sectional curvature is affinely equivalent with either (x 1 2 ±x 2 2 )(x 3 2 ±x 4 2 ...(x 2−1 2 ±x 2 2 ) = 1 or (x 1 2 ±x 2 2 (x 3 2 ±x 4 2 )...(x 2−1 2 ±x 2 2 )x 2m+1 = 1 where the dimensionn satisfiesn = 2m orn =2m + 1. This conjecture was proved in [11] in case the metric is positive definite and in [2] in case the metric is Lorentzian.

The first author introduced a Riemannian invariant denoted by δ and proved in [4] that everyn-dimensional submanifold of the complex hyperbolicm-space ℂHm(4c) of constant holomorphic sectional curvature 4c

We study ane hyperspheres M with constant sectional curvature (with respect to the ane metric h). A conjecture by M. Magid and P. Ryan states that every such ane hypersphere with nonzero Pick invariant is anely equivalent to either where the dimension n satises n =2 m 1o rn =2 m .U p to now, this conjecture was proved if M is positive denite or if M is a 3-dimensional Lorentz space. In this paper, we give an armative answer to this conjecture for arbitrary dimensional Lorentzian ane hyperspheres.

In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold M, denoted respectively by 6(nl,... ,nk) and 5(ni,... ,nk), which trivially satisfy 6(ni,... ,nk) > 6(n,... , nk). In this article, we completely determine the Riemannian manifolds satisfying the condition 6(nfl,... , nk) = S(nl,... ,nk). By applying the notions of these 6-invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.

We study Lagrangian immersions \(M^n_1\) into Lorentzian complex indefinite space forms \(\tilde M^n_1(4 \tilde c), \tilde c \ne 0\) and classify all such immersions which are minimal and have constant curvature \(c \ne \tilde c\).

In analogy to an inequality of Chen [2], Scharlach and co-workers [7] have found a new, optimal inequality for (equi-) affine spheres. We classify those three-dimensional hyperbolic affine spheres for which the corresponding equality is assumed. This complements the classification of the elliptic case [3].

In this paper, we study Lagrangian submanifolds of the nearly Kahler 6-sphere S6(1). We obtain a classification of the Lagrangian submanifolds which admit a unit length Killing vector field whose integral curves are great circles by relating them to almost complex surfaces in S6(1) or holomorphic curves in C P2(4).

It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms 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 M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−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 simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(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.

We prove the pointwise inequality 0 ≥ρ+ρ⊥– 1 involving the normalized scalar curvatureρand normal scalar curvatureρ⊥of a totally real 3-dimensional submanifold of the nearly Kaehler 6-sphere. Further we classify submanifolds realizing the equality in this inequality.

For (positive) definite surfaces in R 4 \mathbb {R}^{4} there is a canonical choice of a centroaffine normal plane bundle, which induces a centroaffine invariant Ricci-symmetric connection ∇ \nabla . We classify all surfaces in R 4 \mathbb {R}^{4} with planar ∇ \nabla -geodesics. It turns out that the resulting class of surfaces is umbilical with projectively flat induced connection and flat normal plane bundle.

We study affine hypersurfaces M which have isotropic difference tensor. Note that any surface always has isotropic difference tensor. Therefore, we may assume that n > 2 . Such hypersurfaces have previously been studied by the first author and M. Djoric in [1] under the additional assumption that M is an affine hypersphere. Here we study the general case. As for affine spheres, we first show that isotropic affine hypersurfaces which are not congruent to quadrics are necessarily 5, 8, 14 or 26 dimensional. From this, we also obtain a complete classification in dimension 5.

A slant immersion is an isometric immersion from a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this paper we establish the existence and uniqueness theorem for slant immersions into complex-space-forms. By applying this result, we prove in this paper several existence and nonexistence theorems for slant immersions. In particular, we prove the existence theorems for slant surfaces with prescribed mean curvature or with prescribed Gaussian curvature. We also prove the non-existence theorem for flat minimal proper slant surfaces in non-flat complex space forms.

Surfaces which are both affine and Euclidean minimal are called Thomsen surfaces. In ℝ3, these surfaces have been completely classified byBarthel, Volkmer andHaubitz. A similar problem, in the Lorentzian 3-space was solved byMagid. In the present paper, we study Thomsen surfaces in ℝ4 and show that these surfaces are affine equivalent to the complex paraboloid.

In [3], B. Y. Chen proved that, for any Lagrangian submanifold M in a complex space-form Mn(4c) (c = ± 1), the squared mean curvature and the scalar curvature of M satisfy the following inequality:He then introduced three families of Riemannian n-manifolds and two exceptional n-spaces Fn, Ln and proved the existence of a Lagrangian isometric immersion pa from into ℂPn(4) and the existence of Lagrangian isometric immersions f, l, ca, da from Fn, Ln, , into ℂHn(− 4), respectively, which satisfy the equality case of the inequality. He also proved that, beside the totally geodesie ones, these are the only Lagrangian submanifolds in ℂPn(4) and in ℂHn(− 4) which satisfy this basic equality. In this article, we obtain the explicit expressions of these Lagrangian immersions. As an application, we obtain new Lagrangian immersions of the topological n-sphere into ℂPn(4) and ℂHn(−4).

In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.