This paper investigates the geometric and topological intricacies involved with several very specialized categories of submanifolds within two particular types of almost contact manifolds: namely, slant and semi-slant submanifolds within generalized Sasakian-space-forms and trans-Sasakian manifolds. Throughout this review, \((F,\;\xi,\;\eta, g)\) will be a metric almost contact structure on a given almost contact manifold \(\tilde M,\) that is, \(F\) is an endomorphism of \(T\tilde M\), \(\xi\) a vector field on \(\tilde M\), \(\eta\) a 1-form on \(\tilde M\) and \(g\) a Riemannian metric on \(\tilde M\) such that \[ F^2=-I+\eta\otimes\xi,\;\eta(\xi)=1,\;g(FX,FY)=g(X,Y)-\eta(X)\eta(Y). \]
Recall that a generalized Sasakian-space-form is an almost contact metric manifold for which the Riemannian curvature operator \(R_{XY}Z\) can be written as an element of the linear span (over the ring of global functions on the manifold) of the \((3,1)\)-tensors \[ \lambda_1(X,Y,Z)=g(Y,Z)X-g(X,Z)Y, \]
\[ \lambda_2(X,Y,Z)=g(Y,FZ)FX-g(X,FZ)FY-2g(X,FY)FZ, \]
\[ \lambda_3(X,Y,Z)=\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X+(g(X,Z)\eta(Y)-g(Y,Z)\eta(X))\xi. \]
When \(R\) can be represented as the span of \(\lambda_1,\;\lambda_2,\;\lambda_3\) with certain constant coefficients, the resulting almost contact structure is either a Sasakian-space-form or a Kenmotsu space-form. Related to this generalization of well-behaved metric contact structures is that of trans-Sasakian manifolds, introducted by Oubina in [J. A. Oubina, Publ. Math. 32, 187–193 (1985; Zbl 0611.53032)]. An \((\alpha,\;\beta)\)-trans-Sasakian manifold is an almost contact manifold on which exist functions \(\alpha\) and \(\beta\) such that \[ \left(\tilde \nabla_X F\right)Y=\alpha\left(g(X,Y)\xi-\eta(Y)X\right)+\beta\left( g(FX,Y)\xi-\eta(Y)FX\right). \] This is another method of generalizing the well-behaved aspects of Sasakian manifolds, first introduced by P. Alegre, D. E. Blair and A. Carriazo, [Isr. J. Math. 141, 157–183 (2004; Zbl 1064.53026)]. It should be noted that an alternative approach – and one that runs more parallel with Oubina’s concepts – is that of the \((\kappa, \mu)\)-nullity condition, which was introduced by D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, [Isr. J. Math. 91, No. 1-3, 189–214 (1995; Zbl 0837.53038)].
In general, very little can be said confidently about general types of submanifolds within almost contact manifolds unless substantial structural restrictions are given on the ambient manifolds or the submanifolds. In this case, the author continues the work begun in [Bull. Transilv. Univ. Brasov Ser. III, 1(50), 79–86 (2008)] and explores the geometry and topology of so-called slant and semi-slant submanifolds within trans-Sasakian manifolds and generalized Sasakian-space-forms. A slant submanifold is a submanifold \(M\subset \tilde M,\) tangent to \(\xi\), such that each unit vector of \(F(D)\) makes a constant angle with \(TM\), where \(D\) is the distribution of \(TM\) orthogonal to \(\xi.\) The constant angle \(\theta\) is called the slant angle of \(M\). This concept originated with Lotta’s work in [A. Lotta, Bull. Math. Sco. Sci. Math. Roum., Nouv. Sér. 39, No. 1-4, 183–198 (1996; Zbl 0885.53058)] and forms an almost contact analogue to slant submanifolds of almost Hermitian manifolds, which themselves are higher-dimensional analogues of loxodromic curves on the sphere.
A semi-slant submanifold is a submanifold \(M\subset \tilde M\) with an orthogonal splitting \(TM=D_1\oplus D_2\oplus \left< \xi\right>\) such that 1) \(F(D_1)=D_1\) and 2) \(D_2\) is slant with slant angle \(\theta.\) These form an obvious generalization of slant submanifolds and were introduced by J. L. Cabrerizo, A. Carriazo, L.M. Fernández and M. Fernández, [Geom. Dedicata 78, No. 2, 183–199 (1999; Zbl 0944.53028)].
Here, the author looks at three different possible configurations of these spaces. First, geometric aspects of semi-slant submanifolds within trans-Sasakian manifolds are explored. More specifically, structural equations for the covariant derivatives of various linear transformations on \(TM\), namely, the projections of \(F\) on \(TM\), are derived using both the Levi-Civita connection of \(g\) and the Bott connection of the splitting \(D_1\oplus D_2\). It is found then that \(g\) is Bott parallel if and only if \(D_1\) is a totally geodesic plane field. Furthermore, a volume form on \(D_1\) that is Bott parallel is constructed. Using these results, the author finds that, if \(M\) is compact, the first \(q\) Betti numbers of \(M\) are nontrivial, where \(q\) is the dimension of \(D_1.\)
In the next section, the author concentrates on the volume stability of compact slant submanifolds in generalized Sasakian space-forms. In particular, conditions for the first and second derivatives of the volume of the submanifold in an orthogonal direction are given for Legendrian, Hamiltonian and harmonic variations, i.e., orthogonal directions that act as infinitesimal Legendrian, Hamiltonian and harmonic flows. A very elegant equation is derived for a partial Ricci tensor of the submanifold: \[ \Sigma_{a=1}^n \tilde R(X,e_a,X,e_a)=nf_1 - f_3, \] for all unit vector \(X\) orthogonal to \(TM\) where \(\{e_1,\dots, e_n\}\) is an orthonormal basis of \(TM\) and \(\tilde R = f_1\lambda_1+ f_2\lambda_2+ f_3\lambda_3.\) This is used to show that, when \(X\), normal to \(TM\) in \(\tilde M,\) is parallel to the Levi-Civita connection of \(\tilde g\), the volume is stable in direction of \(X.\) This is the weakest of the sections, since most of the proofs are not given but simply taken to be “similar arguments” to that within [G. Pitiş, Kyungpook Math. J. 41, No. 2, 381–392 (2001; Zbl 1006.53054)]. This is problematic, since there seems to be a plethora of typographical errors within this section, making independent correction by the reader very difficult.
Lastly, the author looks at the Chern classes of integral submanifolds of manifolds that are both trans-Sasakian and generalized Sasakian-space-forms. By these are meant the Chern classes of the maximal normal bundle of the submanifold that is invariant by \(F.\) This has a natural structure as a complex vector bundle, and conditions under which the first Chern class of this bundle vanishes, are given, namely, when the mean curvature vector of \(M\) is parallel and when \(M\) is a parallel, totally umbilical submanifold.
This is a very interesting paper, typographical errors notwithstanding, that makes a good contribution to a burgeoning field. It serves as a good companion piece to another recent paper [V. A. Khan and M. A. Khan, Sarajevo J. Math. 2(14), 83–93 (2006; Zbl 1126.53030)] on the same topic.