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We provide explicit descriptions of all totally geodesic surfaces of a complex quadric of arbitrary dimension. Totally geodesic submanifolds of complex quadrics were first studied by Chen and Nagano in 1977 and fully classified by Klein in 2008. In particular, we interpret some of these surfaces as Gaussian images of surfaces in a unit three-sphere and all others as elements of the Veronese sequence introduced by Bolton, Jensen, Rigoli and Woodward. We also briefly discuss how the classification can be translated to the non-compact dual of the complex quadric, namely the hyperbolic complex quadric.

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

We study almost complex submanifolds of pseudo nearly Kahler manifolds. We show in particular that a 6 dimensional strict nearly Kahler manifold does not admit any 4 dimensional almost complex submanifolds. This generalises results obtained by Gray (Proc Am Math Soc 20:277–279, 1969) for the nearly Kahler 6-sphere and by Podesta and Spiro (J Geom Phys 60:156–164, 2010) in the Riemannian case.

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions, respectively all but one, coincide.

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space ℝn+1 which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3- and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.

It was proved in Chen and Dillen (J Math Anal Appl 379(1), 229–239, 2011) and Chen et al. (Differ Geom Appl 31(6), 808–819, 2013) that every Lagrangian submanifold M of a complex space form $$\tilde{M}^{n}(4c)$$ with constant holomorphic sectional curvature 4c satisfies the following optimal inequality: A $$\begin{aligned} \delta (2,\ldots ,2)\le \frac{n^2(2n-k-2)}{2(2n-k+4)} H^{2} +\frac{n^2-n-2k}{2}c, \end{aligned}$$ where $$H^{2}$$ is the squared mean curvature and $$\delta (2,\dots ,2)$$ is a $$\delta $$ -invariant on M introduced by the first author, and k is the multiplicity of 2 in $$\delta (2,\dots ,2)$$ , where $$n\ge 2k +1$$ . This optimal inequality improves an earlier inequality obtained by the first author in Chen (Jpn J Math 26(1), 105–127, 2000). The main purpose of this paper is to study Lagrangian submanifolds of $$\tilde{M}^{n}(4c)$$ satisfying the equality case of the optimal inequality (A).

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 $${\mathscr {D}}_1^\perp $$ .

We study Lagrangian submanifolds of the nearly Kahler S 3 × S 3 with respect to their so called angle functions. We show that if all angle functions are constant, then the submanifold is either totally geodesic or has constant sectional curvature and there is a classification theorem that follows from Dioos et al. (2018). Moreover, we show that if precisely one angle function is constant, then it must be equal to 0 , π 3 or 2 π 3 . Using then two remarkable constructions together with the classification of Lagrangian submanifolds of which the first component has nowhere maximal rank from, Bektas et al. (2018), we obtain a classification of such Lagrangian submanifolds.

We show how the metric, the almost complex structure and the almost product structure of the homogeneous nearly Kahler S3 ? S3 can be recovered from a submersion ? : S3 ? S3 ? S3 ? S3 ? S3. On S3 ? S3 ? S3 we have the maps obtained either by changing two coordinates, or by cyclic permutations. We show that these maps project to maps from S3 ? S3 to S3 ? S3 and we investigate their behavior.

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ can only be $$\frac{2}{\sqrt{3}}$$ or $$\frac{4}{\sqrt{3}}$$ .

© 2017, Springer Science+Business Media Dordrecht. We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations. ispartof: Annals of Global Analysis and Geometry vol:52 issue:1 pages:45-55 status: published

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding problem of trying to generalize the classical theorem of Blaschke and Deicke on affine hyperspheres in equiaffine differential geometry to that in centroaffine differential geometry., 14 pages

Copyright © Royal Society of Edinburgh 2018. We study non-Totally geodesic Lagrangian submanifolds of the nearly Kähler 3 × 3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 3. Indeed starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way. ispartof: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS vol:149 issue:3 pages:655-689 status: published

In this paper, through making careful analysis of Gauss and Codazzi equations, we prove that four dimensional biharmonic hypersurfaces in nonzero space forms have constant mean curvature. Our result gives the positive answer to the conjecture proposed by Balmus–Montaldo–Oniciuc in 2008 for four dimensional hypersurfaces.

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

We define two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kahler manifold . As a consequence we can construct sequences of H-surfaces and almost complex surfaces.

© 2015 Elsevier B.V. We give a complete classification of Lagrangian immersions of homogeneous 3-manifolds (the Berger spheres, the Heisenberg group Nil3, the universal covering of the Lie group PSL(2,R) and the Lie group Sol3) in 3-dimensional complex space forms. As a corollary, we get a new characterization of the Berger sphere in complex projective space. publisher: Elsevier articletitle: A new characterization of the Berger sphere in complex projective space journaltitle: Journal of Geometry and Physics articlelink: http://dx.doi.org/10.1016/j.geomphys.2015.02.009 content_type: article copyright: Copyright © 2015 Elsevier B.V. All rights reserved. ispartof: Journal of Geometry and Physics vol:92 pages:129-139 status: published

The analogue of CR-submanifolds in (almost) Kahlerian manifolds is the concept of contact CR-submanifolds in Sasakian manifolds. These are submanifolds for which the structure vector field ξ is tangent to the submanifold and for which the tangent bundle of M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant with respect to the endomorphism φ and E(M) is antiinvariant with respect to φ. The lowest possible dimension for M in which this decomposition is non trivial is the dimension 4. In this paper we obtain a complete classification of four-dimensional contact CR-submanifolds in S5(1) and S7(1) for which the second fundamental form restricted to H(M) and E(M) vanishes identically.

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

© 2017, Mathematical Society of the Rep. of China. All rights reserved. The notion of an ideal submanifold was introduced by Chen at the end of the last century. A survey of recent results in this area can be found in his book [9]. Recently, in [10], an optimal collection of Chen’s inequalities was obtained for Lagrangian submanifolds in complex space forms. As shown in [2], these inequalities have an immediate counterpart in Centroaffine differential geometry. Centroaffine hypersurfaces realising the equality in one of these inequalities are called ideal centroaffine hypersurfaces. So far, most results in this area have only been related with 3- and 4-dimensional δ#(2)-ideal Centroaffine hypersurfaces. The purpose of this paper is to classify δ#(2,2)- ideal hypersurfaces of dimension 5 in centroaffine differential geometry. ispartof: Taiwanese Journal of Mathematics vol:21 issue:2 pages:283-304 status: published

© 2016 Elsevier B.V. It is well-known that Vrancken–Li–Simon classified locally strongly convex affine hyperspheres in Rn+1 whose affine metric are of constant sectional curvatures, but on the other side it is still a difficult problem to classify n-dimensional locally strongly convex affine hyperspheres whose affine metrics are Einstein. In this paper, we have solved the problem in case n=4. publisher: Elsevier articletitle: On four-dimensional Einstein affine hyperspheres journaltitle: Differential Geometry and its Applications articlelink: http://dx.doi.org/10.1016/j.difgeo.2016.10.003 content_type: article copyright: © 2016 Elsevier B.V. All rights reserved. ispartof: Differential Geometry and its Applications vol:50 pages:20-33 status: published

In affine differential geometry, Calabi discovered how to associate a new hyperbolic affine hypersphere with two hyperbolic affine hyperspheres. This was later generalized by Dillen and Vrancken in order to obtain a large class of examples of equiaffine homogeneous affine hypersurfaces. Note that the constructions defined above remain valid if one of the affine hyperspheres is a point. In this paper we consider the converse question: how can we determine, given properties of the difference tensor K and the affine shape operator S, whether a given hypersurface can be decomposed as a generalized Calabi product of an affine sphere and a point?

For the memory of Franki Dillen

In his book on Pseudo-Riemannian geometry, δ -invariants and applications, B.Y. Chen introduced a sequence of curvature invariants. Each of these invariants is used to obtain a lower bound for the length of the mean curvature vector for an immersion in a real space form. A submanifold is called an ideal submanifold, for that curvature invariant, if and only if it realizes equality at every point. The first such introduced invariant is called δ ( 2 ) . On the other hand, a well known notion for submanifolds of Sasakian space forms, is the notion of a contact CR-submanifold. In this paper we combine both notions and start the study of minimal contact CR-submanifolds which are δ ( 2 ) ideal. We relate this to a special class of surfaces and obtain a complete classification in arbitrary dimensions.

Let M n be an n-dimensional Lagrangian submanifold of a complex space form M ˜ n ( 4 c ) of constant holomorphic sectional curvature 4c. We prove a pointwise inequality δ ( n 1 , … , n k ) ⩽ a ( n , k , n 1 , … , n k ) ‖ H ‖ 2 + b ( n , k , n 1 , … , n k ) c , with on the left-hand side any delta-invariant of the Riemannian manifold M n and on the right-hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right-hand side are optimal in the sense that there exist non-minimal examples satisfying equality at least one point. We also characterize those Lagrangian submanifolds satisfying equality at any of their points. Our results correct and extend those given in [6] .

© 2016, Springer International Publishing. We study four-dimensional locally strongly convex, locally homogeneous, hypersurfaces whose affine shape operator has two distinct principal curvatures. In case that one of the eigenvalues has dimension 1 these hypersurfaces have been previously studied in Dillen and Vrancken (Math Z 212:61–72, 1993, J Math Soc Jpn 46:477–502, 1994) and Hu et al. (Differ Geom Appl 33:46–74, 2014) in which a classification of such submanifolds was obtained in dimension 4 and 5 under the additional assumption that the multiplicity of one of the eigenvalues is 1. In this paper we complete the classification in dimension 4 by considering the case that the multiplicity of both eigenvalues is 2. ispartof: Journal of Geometry vol:108 issue:1 pages:119-147 status: published

We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct., Comment: 14 pages

Let M n be an n -dimensional ( n ≥ 3 ) minimal Lagrangian isotropic submanifold in an indefinite complex space form. We show that the dimension of M n satisfies n = 3 r + 2 with r , a positive integer. When n 14 , we give a classification of such submanifolds.

We consider minimal closed hypersurfaces \({M \subset \mathbb{S}^7(1)}\) with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.

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.

In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric.

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.

A six-dimensional unit sphere has an almost complex structure J defined by the vector cross product on the space of purely imaginary Cayley numbers, which makes S 6 a nearly Kähler manifold. In this paper, we study 3-dimensional CR submanifolds of S 6(1), investigating certain geometric conditions. We show that if such a submanifold attains equality in Chen's inequality, it is always minimal. We recall that a classification of minimal 3-dimensional submanifolds was obtained in [Djorić, Vrancken, J. Geom. Phys. 56: 2279–2288, 2006]. For 3-dimensional CR submanifolds, the restriction of the almost complex structure J to the tangent space automatically induces an almost contact structure on the submanifold. We prove that this structure is not Sasakian with respect to the induced metric. We also give an example from [Hashimoto, Mashimo, J. Math. 28: 579–591, 2005], see also [Ejiri, Trans. Amer. Math. Soc. 297: 105–124, 1986], of a tube around a superminimal almost complex curve in S 6(1) for which this almost contact structure is Sasakian with respect to a constant scalar multiple of the induced metric.

In this paper, we study totally real submanifolds of the nearly Kahler six-dimensional unit sphere. Since in this case also, parallel submanifolds are totally geodesic, we introduce a weaker condition, namely that for any tangent vector v 〈(∇h)(v,v,v),Jv〉=0. We obtain a complete classification of totally real three-dimensional submanifolds of S6(1) satisfying the above condition.

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.

Starting from two Lagrangian immersions and a horizontal curve in S3(1), it is possible to construct a new Lagrangian immersion, which we call a warped-product Lagrangian immersion. In this paper, we find two characterizations of warped-product Lagrangian immersions. We also investigate Lagrangian submanifolds which attain at every point equality in the improved version of Chen's inequality for Lagrangian submanifolds of ℂPn(4) as discovered by Opreaffi We show that, for n≥4, an n-dimensional Lagrangian submanifold in ℂPn(4) for which equality is attained at all points is necessarily minimal.

In this paper, we study certain three-dimensional CR-submanifolds M of the nearly Kähler 6-dimensional sphere S6(1). It is well known that there does not exist a three-dimensional totally geodesic proper CR-submanifold in S6(1). In this paper we obtain a classification of the 3-dimensional CR-submanifolds which are the closest possible to totally geodesic submanifolds, i.e. those that admit a one-dimensional nullity distribution.