- Open Access
On Chen invariants and inequalities in quaternionic geometry
© Vîlcu; licensee Springer 2013
- Received: 26 September 2012
- Accepted: 9 February 2013
- Published: 22 February 2013
In this paper we give a survey of main results concerning Chen inequalities for submanifolds in quaternionic space forms and propose some open problems in the field for further research.
AMS Subject Classification:53C15, 53C25, 53C40.
- Chen’s invariant
- squared mean curvature
- shape operator
- quaternionic space form
- slant submanifold
The theory of Chen invariants, initiated by Prof. B.-Y. Chen in a seminal paper published in 1993 , is presently one of the most interesting research topic in differential geometry of submanifolds. The author’s original motivation to introduce new types of Riemannian invariants, known as δ-invariants or Chen invariants, was the need to provide answers to an open question raised by Chern concerning the existence of minimal immersions into a Euclidean space of arbitrary dimension . In fact, due the lack of control of the extrinsic properties of the submanifolds by the known intrinsic invariants, no solutions to Chern’s problem were known before the invention of Chen invariants (see ). Therefore, in , Chen obtained a necessary condition for the existence of minimal isometric immersion from a given Riemannian manifold M into Euclidean space and established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature (both being intrinsic invariants) and squared mean curvature (the main extrinsic invariant). On the other hand, in , Chen obtained inequalities between the k-Ricci curvature, the squared mean curvature and the shape operator for submanifolds in real space forms with arbitrary codimensions. These inequalities are also sharp, and many nice classes of submanifolds realize equality in all above inequalities. Since then many papers concerning Chen invariants and inequalities have appeared in the literature for different classes of submanifolds in various ambient spaces, like complex space forms [5–7], cosymplectic space forms [8–10], Sasakian space forms [11–13], locally conformal Kähler space forms [14–16], generalized complex space forms [17–19], locally conformal almost cosymplectic manifolds [20, 21], -contact space forms [22, 23], Kenmotsu space forms [24, 25], S-space forms [26, 27], T-space forms ; see also  and references therein.
In the last decade, Chen-like inequalities were extended in the quaternionic setting. It is the main purpose of this paper to present the evolution of these inequalities for submanifolds in quaternionic space forms, to survey some recent results on this topic and to propose a set of natural problems in the field.
2.1 Riemannian invariants
In this subsection we recall some basic concepts concerning Chen invariants, using mainly .
For an integer , we denote by the set of k-tuples of integers ≥2 satisfying , . We denote by the set of unordered k-tuples with for a fixed n.
We note that such a curvature is called a k-Ricci curvature.
where L runs over all k-plane sections in and X runs over all unit vectors in L.
It is well known that all these above invariants have many interesting applications to several fields of mathematics (see ).
2.2 Quaternionic Kähler manifolds
The geometry of Riemannian structures, complex structures, almost contact structures, hypercomplex structures, quaternionic and Cauchy-Riemann structures belongs to the general theory of the G-structures, a domain of great interest in modern differential geometry, in global analysis and mathematical physics. Quaternionic manifolds correspond to the reduction of the structural group at . From the metric viewpoint, the most interesting is the case of quaternionic Kähler manifolds, which corresponds to the reduction of the holonomy at a subgroup of . They appear in Berger’s list for possible holonomy groups of irreducible Riemannian manifolds . We give in this subsection a quick review of basic definitions and properties concerning the differential geometry of manifolds endowed with quaternionic structures. For details, see .
where Id denotes the identity tensor field of type on M and the indices are taken from modulo 3. Then the bundle σ is called an almost quaternionic structure on M and is called a canonical local basis of σ. Moreover, is said to be an almost quaternionic manifold. It is easy to see that any almost quaternionic manifold is of dimension 4m, .
for all vector fields X, Y on and any canonical local basis of σ. Moreover, is said to be an almost quaternionic Hermitian manifold.
for any vector field X on , where the indices are taken from modulo 3.
for all vector fields X, Y, Z on and any local basis of σ. We note that some interesting characterizations of quaternionic space forms were obtained in .
for all and .
The submanifold M is called totally geodesic if the second fundamental form vanishes identically and totally umbilical if there is a real number λ such that for any tangent vectors X, Y on M. If , then the submanifold M is said to be minimal.
The submanifold M is said to satisfy Chen’s equality if the equality case of (15) holds identically.
In quaternionic Kähler ambient, the first classes of submanifolds which have been introduced and studied were quaternionic  and totally real submanifolds . A submanifold M in a quaternionic Kähler manifold is called a quaternionic submanifold (resp. a totally real submanifold) if each tangent space of M is carried into itself (resp. into the normal space) by each section in σ. An n-dimensional totally real submanifold of a quaternionic space form is said to be a Lagrangian submanifold if . It is well known that a quaternionic submanifold is totally geodesic, so the first class of interest from Chen’s inequalities viewpoint is given by the totally real submanifolds. In , the validity of the inequality (15) was proved for totally real submanifolds in a quaternionic space form of quaternionic sectional curvature 4c, and Chen’s equality was interpreted in terms of eigenvalues and eigenspaces of the Weingarten operators of the submanifold. As a consequence, Hong and Houh obtained the following interesting result.
Theorem 3.1 
Every Lagrangian submanifold of a quaternionic space form satisfying Chen’s equality is minimal.
In , Şahin introduced the concept of slant submanifolds as a natural generalization of both quaternionic and totally real submanifolds. A submanifold M of a quaternionic Kähler manifold is said to be a slant submanifold if for each non-zero vector X tangent to M at p, the angle between and , is constant, i.e., it does not depend on choice of and . We can easily see that quaternionic submanifolds are slant submanifolds with and totally-real submanifolds are slant submanifolds with . A slant submanifold of a quaternionic Käler manifold is said to be proper (or θ-slant proper) if it is neither quaternionic nor totally real.
In , the present author obtained the generalization of the first Chen inequality to the case of slant submanifolds in quaternionic space forms as follows.
Theorem 3.2 
for any k-tuples . Immersion of a submanifold which realizes equality in this inequality at every point is said to be an ideal immersion.
In , Yoon proved that the inequality (19) is also true for totally real submanifolds in a quaternionic space forms of quaternionic sectional curvature 4c. This inequality was later generalized for slant submanifolds in quaternionic space forms as follows.
Theorem 3.3 
for any k-tuples .
which is known as the Chen-Ricci inequality. In , Liu obtained the same inequality for totally real submanifolds in quaternionic space forms.
A submanifold M of a quaternion Kähler manifold is said to be a quaternionic CR-submanifold if there exists two orthogonal complementary distributions D and on M such that D is invariant under quaternionic structure and is totally real (see ). An estimation of the Ricci curvature of a quaternionic CR-submanifold in a quaternionic space form has been established in , as follows.
Theorem 4.1 
Let M be an n-dimensional quaternionic CR-submanifold of a quaternionic space form . Then:
- (ii)If , then a unit tangent vector X at p satisfies the equality case of (24) (respectively (25)) if and only if (respectively ), where is the relative null space of M at the point defined by
We note that an optimal inequality concerning the Ricci curvature for quaternionic CR-submanifolds of quaternionic space forms with a semi-symmetric metric connection was obtained in . It is clear that, although quaternionic CR-submanifolds are also the generalization of both quaternionic and totally real submanifolds, there exists no inclusion between the two classes of quaternionic CR-submanifolds and slant submanifolds. The following estimation of the Ricci curvature for slant submanifolds in quaternionic space forms was firstly proved in , and an alternative nice proof can be found in .
Theorem 4.2 
- (i)For each unit vector , we have(26)
If , then a unit tangent vector X at p satisfies the equality case of (26) if and only if X belongs to the relative null space of M at p.
The equality case of (26) holds identically for all unit tangent vectors at p if and only either p is a totally geodesic point or and p is a totally umbilical point.
In , Oprea proved using optimization methods that the inequality (23) is not optimal for Lagrangian submanifolds in complex space forms and obtained an improved Chen-Ricci inequality. In , Deng obtained the proof of the improved inequality as an application of suitable algebraic inequalities. This allows one to determine the equality condition in an explicit form. Recently, in , the Chen-Ricci inequality was improved for Lagrangian submanifolds in quaternionic space forms as follows.
Theorem 4.3 
p is a totally geodesic point or
- (ii)and p is an H-umbilical point with , , i.e., there exists an orthonormal basis of such that the second fundamental form takes the following simple form:
for some suitable functions and satisfying , .
at p, where denotes the identity map of .
The inequality (28) was extended in the setting of a totally real submanifold in a quaternionic space form by Liu and Dai in  and recently, in , both inequalities (28) and (29) were generalized for proper slant submanifolds in quaternionic space forms as follows.
Theorem 5.1 
Theorem 5.2 
- (i)If , then the shape operator at the mean curvature satisfies(31)
If , then at p.
- (iii)A unit vector satisfies(32)
- (iv)The identity(33)
holds at p if and only if p is a totally geodesic point.
In this section we propose to investigate the following open problems related to the Chen invariants and ideal immersions in quaternionic space forms.
Problem 6.1 QR-submanifolds were introduced by Bejancu  as a generalization of the real hypersurfaces of a quaternionic Kähler manifold. In fact, a real submanifold M of a quaternionic Kähler manifold is called a QR-submanifold if there exists a vector subbundle D of the normal bundle such that and , for all , and for any local basis of σ, where is the complementary orthogonal bundle to D in . The problem is to extend the inequalities (15), (19), (23), (28) and (29) for QR-submanifolds in quaternionic space forms.
Problem 6.2 To generalize the Theorems 3.2, 3.3, 4.1, 4.2, 5.1 and 5.2 for semi-slant submanifolds in quaternionic space forms.
We note that the concept of a semi-slant submanifold in quaternionic geometry was introduced by Şahin  as follows: a real submanifold M of a quaternionic Kähler manifold is said to be a semi-slant submanifold if there exist two orthogonal vector subbundles μ and of the normal bundle such that , is anti-invariant with respect to and is slant with respect to for all , and for any local basis of σ. It is easy to see that this notion is natural because QR-submanifolds and in particular real hypersurfaces are examples of semi-slant submanifolds of a quaternionic Kähler manifold and therefore fulfill the main purpose for which semi-slant submanifolds were introduced in Kähler geometry by Papaghiuc . We also remark that recently, in , Shukla and Rao defined another concept of a semi-slant submanifold of a quaternionic Kähler manifold by analogy with the definition of Papaghiuc, but that class of submanifolds, although generalizes slant submanifolds, does not contain real hypersurfaces as a subclass.
for all vector fields X, Y on , where the indices are taken from modulo 3 and denotes the tangential component of . The problem is to extend the improved Chen-Ricci inequality (27) to quaternionic slant submanifolds in quaternionic space forms and to investigate the equality case of the inequality.
Problem 6.4 In , Oprea improved the inequality (15) for Lagrangian submanifolds in complex space forms and recently, in , Chen and Dillen obtained improved general inequalities which involve the squared mean curvature and the invariants for Lagrangian submanifolds in complex space forms, giving also necessary and sufficient condition for a Lagrangian submanifold to attain the equality for arbitrary . The problem is to obtain improved general inequalities for Lagrangian submanifolds in quaternionic space forms and to completely classify Lagrangian submanifolds which realize the equality case of these inequalities.
Problem 6.5 In  it was proved that there do not exist quaternionic slant immersions of minimal codimension in quaternionic projective space with unfull first normal bundle. An interesting problem is to investigate the existence of quaternionic slant immersions of minimal codimension in a quaternionic projective space satisfying the equality case of (19) such that either or and at least one of is 2. We note that the answer of the corresponding problem in Kählerian geometry is negative .
Problem 6.6 To completely classify -ideal slant submanifolds and -ideal Lagrangian submanifolds in quaternionic space forms.
Problem 6.7 To obtain Chen-like inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms and to completely classify Casorati ideal submanifolds.
It is known that the Casorati curvature of a submanifold in a Riemannian manifold is an extrinsic invariant defined as the normalized square of the length of the second fundamental form. Moreover, this notion extends the concept of the principal direction of a hypersurface of a Riemannian manifold to submanifolds of a Riemannian manifold and it was preferred by Casorati over the traditional Gauss curvature because corresponds better with the common intuition of curvature . Therefore it is of great interest to obtain optimal inequalities for the Casorati curvatures of submanifolds in different ambient spaces. We note that in , Decu, Haesen and Verstraelen obtained some optimal inequalities involving the scalar curvature and the Casorati curvature of a Riemannian submanifold in a real space form and the holomorphic sectional curvature and the Casorati curvature of a Kähler hypersurface in a complex space form. Moreover, the same authors proved in  an inequality in which the scalar curvature is estimated from above by the normalized Casorati curvatures.
This paper is a contribution to the International Conference on Applied Analysis and Algebra (ICAAA2012) (20-24 June 2012, Istanbul, Turkey).
Dedicated to Prof. Ravi P Agarwal on his 65th birth anniversary.
This work was supported by CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0118.
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