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ispartof: RESULTS IN MATHEMATICS vol:77 issue:3 status: published

It is known that there exist only four six-dimensional homogeneous non-Kahler, nearly Kahler manifolds: the sphere $$\mathbb {S}^6$$ , the complex projective space $${\mathbb {C}P}^3$$ , the flag manifold $${\mathbb {F}}^3$$ and $$\mathbb {S}^3\times \mathbb {S}^3$$ . So far, most of the results about submanifolds have been obtained when the ambient space is the nearly Kahler $$\mathbb {S}^6$$ . Recently, the investigation of almost complex and Lagrangian submanifolds of the nearly Kahler $$\mathbb {S}^3\times \mathbb {S}^3$$ has been initiated. Here we start the investigation of three-dimensional CR submanifolds of $$\mathbb {S}^3\times \mathbb {S}^3$$ . The tangent space of three-dimensional CR submanifold can be naturally split into two distributions $$\mathscr {D}_1$$ and $${\mathscr {D}}_1^\perp $$ . In this paper, we found conditions that three-dimensional CR submanifolds with integrable almost complex distribution $$\mathscr {D}_1$$ should satisfy, and we give some constructions which allow us to define a wide-range family of examples of this type of submanifolds. Our main result is classification of the three-dimensional CR submanifolds with totally geodesics both, almost complex distribution $$\mathscr {D}_1$$ and totally real distribution $${\mathscr {D}}_1^\perp $$ .

By employing a nice adapted frame we prove a Bonnet-type existence and uniqueness theorem for almost complex surfaces in the homogeneous nearly Kahler manifold $$S^3\times S^3$$ . The proof uses a local correspondence between almost complex surfaces in $$S^3\times S^3$$ and surfaces in $$\mathbb {R}^3$$ that satisfy the Wente H-surface equation. Furthermore we give a complete classification of flat almost complex surfaces in the homogeneous nearly Kahler $$S^3\times S^3$$ .

© 2015, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg. In this paper, continuing with Hu–Li–Vrancken and the recent work of Antić–Dillen- Schoels–Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex affine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S. ispartof: Acta Mathematica Sinica vol:31 issue:10 pages:1531-1554 status: published

It was proven in [B.-Y. Chen, F. Dillen, J. Van der Veken and L. Vrancken, Curvature inequalities for Lagrangian submanifolds: the final solution, Differ. Geom. Appl. 31 (2013), 808-819] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{n}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: \begin{align*} \delta(2,n-2) \leq \frac{n^2(n-2)}{4(n-1)} H^2 + 2(n-2) c, \end{align*} where $H^2$ is the squared mean curvature and $\delta(2,n-2)$ is a $\delta$-invariant on $M$. In this paper we classify Lagrangian submanifolds of complex space forms $\tilde M^{n}(4c)$, $n \geq 5$, which satisfy the equality case of this inequality at every point., Comment: 26 pages

For the memory of Franki Dillen

Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. Recently, it was proved in Chen and Dillen (2011) [11] that for any Lagrangian submanifold M of a complex space form M ˜ n ( 4 c ) , n ⩾ 3 , of constant holomorphic sectional curvature 4c we have δ ( n − 1 ) ⩽ n − 1 4 ( n H 2 + 4 c ) , where H 2 is the squared mean curvature and δ ( n − 1 ) is a δ-invariant of M (cf. Chen, 2000, 2011 [7] , [10] ). In this paper, we completely classify non-minimal Lagrangian submanifolds of complex space forms M ˜ n ( 4 c ) , c = 0 , 1 , − 1 , which satisfy the equality case of the inequality identically.

We study Lorentzian affine hypersurfaces of $${\mathbb{R}^{n+1}}$$ having parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric. As main result, we obtain a complete classification of these hypersurfaces.

On a smooth manifold $M$ we introduce the concept of Codazzi-equivalent Riemannian metrics. The curvature tensors of two Codazzi-equivalent metrics satisfy a simple relation. The results together with known facts about Codazzi tensors give a method of proof for old and new local and global uniqueness results for Riemannian manifolds and Euclidean hypersurfaces.

There exists a well known construction which allows to associate with two Lagrangian immersions \(f_{i} : M_{i} \rightarrow {\mathbb{C}}P^{n_{i}} (4), i = 1, 2\) a family of new Lagrangian immersion from \(I \times M_{1} \times M_{2}\) into \({\mathbb{C}}P^{n_{1}+n_{2}+1} (4)\). Each member of the family is determined by a horizontal curve in S3 (1). In this paper we continue to deal with the inverse problem: how to determine from properties of the second fundamental form whether a given Lagrangian immersion of \(M \rightarrow {\mathbb{C}}P^{n}\) (4) can be locally decomposed in such a way.

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.

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 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.

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.

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.

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.

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).

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.

In this paper, we study 3-dimensional totally real submanifolds of S6(1). If this submanifold is contained in some 5-dimensional totally geodesic S5(1), then we classify such submanifolds in terms of complex curves in CP 2(4) lifted via the Hopf fibration S5(1)→ CP 2(4). We also show that such submanifolds always satisfy Chen’s equality, i.e. δM = 2, where δM (p) = τ(p)−inf K(p) for every p ∈ M . Then we consider 3-dimensional totally real submanifolds which are linearly full in S6(1) and which satisfy Chen’s equality. We classify such submanifolds as tubes of radius π/2 in the direction of the second normal space over an almost complex curve in S6(1).

In this paper we prove that any hypersurface in En+1 of the form where P 1 is a polynomial of degree ≥2 and P 2, ... , P n are functions such that P ′ P ″ = 0 somewhere for all i = 2, ... , n, is of infinite type. As a consequence, we deduce that a polynomial translation hypersurface in En+1, i. e. a hypersurface of the above form where P 1, ... , P n are polynomials, is of finite type if and only if it is a hyperplane. This provides some partial solutions to a problem of B. Y. Chen [C3].

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: Mn → Rn+1 be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of Mn into Rn + 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.

We study affine surfaces which are both affine maximal and affine harmonic. We prove that an indefinite surface satisfying both conditions is affine equivalent to an open part \((u,{1\over 2}u^2,P_1(u)+\upsilon,P_2(u)+{1\over 2}\upsilon^2)\), where P1 and P2 are arbitrary functions of one variable.

In this paper we study the affine Gauss maps of nondegenerate surfaces in R 4 {\mathbb {R}^4} with respect to the Burstin-Mayer normalization (1927) and with respect to the normalization obtained by Nomizu and the first author in 1993. We determine up to affine transformations all surfaces in R 4 {\mathbb {R}^4} with constant affine Gauss maps.

We show how a ruled minimal Lagrangian submanifold of complex projective 3-space may be used to construct two related minimal surfaces in the 5-sphere.

We study minimal Lagrangian immersions from an indefinite real space form M s n ( c ) M^n_s(c) into an indefinite complex space form M ~ s n ( 4 c ~ ) {\tilde {\mathbb {M}}}^n_s(4\tilde c) . Provided that c ≠ c ~ c \ne \tilde c , we show that M s n ( c ) M^n_s(c) has to be flat and we obtain an explicit description of the immersion. In the case when the metric is positive definite or Lorentzian, this result was respectively obtained by Ejiri (1982) and by Kriele and the author (1999). In the case that c = c ~ c = \tilde c , this theorem is no longer true; see for instance the examples discovered by Chen and the author (accepted for publication in the Tôhoku Mathematical Journal).

In this paper we study the affine Gauss maps of nondegenerate surfaces in R4 with respect to the Burstin-Mayer normalization (1927) and with respect to the normalization obtained by Nomizu and the first author in 1993. We determine up to affine transformations all surfaces in R4 with constant affine Gauss maps.

Let M M be a compact 3 3 -dimensional totally real submanifold of the nearly Kaehler 6 6 -dimensional unit sphere. Let K K be the sectional curvature function of M M . Then, if K > 1 / 16 K > 1/16 , M M is a totally geodesic submanifold (and K ≡ 1 K \equiv 1 ).