Cengizhan Murathan

Uludag University · Department of Mathematics

The first and second authors introduced Riemannian warped product submersions and discussed interesting fundamental geometric properties of such submersions in Küpeli Erken and Murathan, Results Math 76(1) (2021), https://doi.org/10.1007/s00025-020-01310-4. In the present paper, we extend this study to put light on the curvature properties of such...

Neill [5] and Gray [1] investigated the Riemannian submersion between Riemannian manifolds. These submersions were later extensively studied in differential geometry.

The study of warped products plays versatile roles in differential geometry as well as in mathematical physics, especially in general relativity (GR). In the present paper, we study statistical submanifolds in a statistical warped product with some related examples. For such submanifolds, we establish a Chen's first inequality and also discuss the...

The study of warped products plays versatile roles in differential geometry as well as in mathematical physics, especially in general relativity $(GR)$. In the present paper, we study statistical submanifolds in a statistical warped product with some related examples. For such submanifolds, we establish a Chen's first inequality and also discuss th...

In this paper, we introduce Riemannian warped product submersions and construct examples and give fundamental geometric properties of such submersions. On the other hand, a necessary and sufficient condition for a Riemannian warped product submersion to be totally geodesic, totally umbilic and minimal is given.

In this paper, we derive Chen inequality for statistical submanifold of statistical warped product manifolds [Formula: see text]. Further, we derive Chen inequality for Legendrian statistical submanifold in statistical warped product manifolds [Formula: see text]. We also provide some applications of derived inequalities in a statistical warped pro...

In this paper, we construct some general geometric inequalities for statistical submanifold of Kenmotsu statistical manifold of constant φ-sectional curvature. Further, we develop Chen-Ricci inequality and B. Y. Chen inequality for statistical submanifold of same ambient manifold. Some consequences of derived inequalities are also described.

In this paper, we give a neutral relation between metallic structure and almost quadratic metric ϕ-structure. Considering N as a metallic Riemannian manifold, we show that the warped product manifold ℝ × f N has an almost quadratic metric ϕ-structure. We define Kenmotsu quadratic metric manifolds, which include cosymplectic quadratic manifolds when...

In this paper, metallic structure and almost quadratic metric phi-structure are studied. Based on metallic (polynomial) Riemannian manifold, Kenmotsu quadratic metric manifold, cosymplectic quadratic metric manifold are defined and gave some examples. Finally, we construct quadratic phi-structure on the hypersurface M^n of a locally metallic Rieman...

Main interest of the present paper is to obtain the generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu manifolds.

In this paper, we study statistical manifolds and their submanifolds. We first construct two new examples of statistical warped product manifolds and give a method how to construct Kenmotsu-like statistical manifold and cosymplectic-like statistical manifold based on the existence of Kaehler-like statistical manifold. Then we obtain the general Win...

We classify three-dimensional paracontact metric manifold whose Ricci operator Q is invariant along Reeb vector field, that is, \({\mathcal {L}} _{\xi }Q=0\).

This paper is a study of almost contact statistical manifolds. Especially this study is focused on almost cosymplectic statistical manifolds. We obtained basic properties of such manifolds. It is proved a characterization theorem and a corollary for the almost cosymplectic statistical manifold with Kaehler leaves. We also study curvature properties...

The purpose of the paper is to prove that if the metric of a 3-dimensional para-Sasakian structure on a semi-Riemannian manifold is a Yamabe soliton then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either (M,g) has constant curvature -1 and reduces to an Einstein manifold, or V is an...

This paper is a study of three-dimensional paracontact metric (Formula presented.)-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field (Formula presented.) is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric (Formula presented.)-manifolds under a condition...

This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly K\"ahler manifold. Finally, we prove that there ex...

We introduce and characterize slant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of slant Riemannian submersions de…ned on Sasakian manifolds. We give a su¢ cient condition for a slant Riemannian submersion from Sasakian manifolds onto Riemannian manifolds to be harmonic. We also give an example o...

The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. We proved the nonexistence of (anti-invariant) Riemannian submersions from Kenm...

We study the Riemann curvature tensor of \((\kappa ,\mu ,\nu )\) -contact metric manifolds, which we prove to be completely determined in dimension 3. We also observe how this curvature tensor is affected by \(D_a\)-homothetic deformations, which will prompt the definition and study of generalized \((\kappa ,\mu ,\nu )\) -space forms and of the nec...

The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. The condition for anti-invariant submersions such that characteristic vector fi...

This paper is a complete study of almost -paracosymplectic manifolds. Basic properties of such manifolds are obtained and general curvature identities are proved. The manifolds with para-Kaehler leaves are characterized. It is proved that, for dimensions greater than , almost -paracosymplectic manifolds are locally conformal to almost paracosymplec...

The purpose of this paper is to study a new class of contact manifolds. Such manifolds are called almost f-cosymplectic manifolds. Several tensor conditions are studied for such type of manifolds. We conclude our results with two examples of almost f-cosymplectic manifolds.

We consider minimal anti-invariant semiparallel submanifolds of generalized Sasakian space forms. We show that the submanifolds are totally geodesic under certain conditions.

This paper is a complete study of almost {\alpha}-paracosmplectic manifolds. We characterize almost {\alpha}-paracosmplectic manifolds which have para Kaehler leaves. Main curvature identities which are fulfilled by any almost {\alpha}-paracosmplectic manifold are found. We also proved that {\xi} is a harmonic vector field if and only if it is an e...

The purpose of this paper is to study a new class of framed manifolds. Such manifolds are called almost α-cosymplectic f-manifolds. For some special cases of α and s, one obtains (almost) α-cosymplectic, (almost) C-manifolds, and (almost) Kenmotsu f-manifolds. Moreover, several tensor conditions are studied. We conclude our results with a general e...

The purpose of this paper is to study a new class of framed manifolds. Such manifolds are Called almost alpha-cosymplectic f-manifolds. For some special cases of alpha and s, one obtains (almost) alpha-cosymplectic, (almost) C-manifolds, and (almost) Kenmotsu f-manifolds. Moreover, several tensor conditions are studied. We conclude our results with...

In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submersions according to characteris...

We give necessary and sufficient conditions for warped product manifolds (M, g), of dimension >= 4, with 1-dimensional base, and in particular, for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R.C-C.R, formed from the curvature tensor R and the Weyl conformal curvature...

In this paper, we characterize three dimensional paracontact metric manifolds whose Reeb vector field {\xi} is harmonic. The paper is also a complete study of 3-dimensional paracontact metric ({\kappa},{\mu},{\nu})-manifolds. We investigated the properties of such manifolds according to the cases {\kappa}>-1, {\kappa}=-1, {\kappa}<-1. Finally examp...

We introduce anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on Sasakian manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of an...

We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give exampl...

The main interest of the present paper is to classify the almost cosymplectic 3-manifolds that satisfy ∥ grad λ∥= const ·(≠0) and ∇ ξ h=2ahϕ.

We introduce slant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of slant Riemannian submersions defined on Sasakian manifolds. We also give an example of such slant submersions.

The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $% \tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was rece...

We give a local classification of ( $\kappa $ , $\mu $ , $\upsilon =const.$ )-contact metric manifold $(M,\phi ,\xi ,\eta ,g)$ with $\kappa <1$ which satisfies the condition “the Boeckx invariant function $I_{M}=\frac{1-\frac{\mu }{2}}{\sqrt{1-\kappa }}$ is constant along the integral curves of the characteristic vector field $\xi $ ”.

We classify the contact metric 3-manifolds that satisfy ||grad{\lambda}||=1 and \nabla_{{\xi}}{\tau}=2a{\tau}{\phi}.

The main interest of the present paper is to prove the dual results for semi-Riemannian submersions, i.e., a semi-Riemannian submersion from a 3-dimensional space form into a surface is biharmonic if and only if it is harmonic. We prove that there is no biharmonic semi-Riemannian submersion from anti-de Sitter space onto a Riemannian manifold. We a...

In this paper, we prove Chen inequalities for submanifolds of a cosym-plectic space form of constant φ-sectional curvature N 2m+1 (c) endowed with a semi-symmetric metric connection, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and sectional curvatures, k-Ricci curvature and the sectional c...

We prove Chen inequalities for submanifolds of a cosymplectic space form of constant φ-sectional curvature N 2m+1 (c) endowed with a semi-symmetric metric connection, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and the sectional curvatures, the k-Ricci curvature and the sectional curvature...

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to...

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c 1 centered at origin with an Euclidean planar curve c 2 has harmonic Gauss map if and only if M is a part of a pl...

In this article we investigate Vranceanu rotation surfaces with pointwise 1- type Gauss map in Euclidean 4-space $ \mathbb{E}^4 $ \mathbb{E}^4 . We show that a Vranceanu rotation surface M has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficent conditions for Vranceanu rotation surface to have pointw...

In this paper we prove Chen inequalities for submanifolds of a lo-cally conformal almost cosymplectic manifold N 2m+1 (c) of constant ϕ-sectional curvature c endowed with a semi-symmetric metric connec-tion, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and sectional curvatures, Ricci curvat...

In this article we investigate Vranceanu rotation surfaces with pointwise 1-type Gauss map in Euclidean 4-space E-4. We show that a Vranceanu rotation surface M has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficent conditions for Vranceanu rotation surface to have pointwise 1-type Gauss map.

The object of the paper is to study some smooth surfaces M whose mean curvature vector H satisfies the H-recurrent condition DX H = λ(X)H in m-dimensional Euclidean space Em, where X is a tangent vector field of M and λ is a 1-form. First of all,we prove that the surfaces which satisfy the Hrecurrent condition in Em are R⊥-parallel (i.e., R⊥H = 0)....

For the Monge-Ampère Z xx Z yy -Z xy 2 =b 20 x 2 +b 11 xy+b 02 y 2 +b 00 we study the question of existence of a solution Z(x,y) in the class of polynomials, where Z(x,y) is the graph of a convex surface. If Z is a polynomial of odd order, a solution does not exist. If Z is a polynomial of fourth order and 4b 20 b 02 -b 11 2 >0, a solution also doe...

We study on a Riemannian manifold (M, g) with a semi-symmetric non-metric connection. We obtain some characterizations for (M, g) satisfying some semisymmety conditions.

For the Monge-Ampere equation ZxxZyy-Z2xy = b20x2+b11xy + b02y2 + 600 we consider the question on the existence of a solution Z(x, y) in the class of polynomials such that Z = Z(x, y) is a graph of a convex surface. If Z is a polynomial of odd degree, then the solution does not exist. If Z is a polynomial of 4-th degree and 4b20b02 - b211 > 0, then...

We consider an anti-invariant, minimal, pseudoparallel and Ricci-generalized pseudoparallel submanifold M of a Kenmotsu space form M (c), for which ξ is tangent to M .

We study pseudo symmetric (briefly ) and pseudo Ricci symmetric (briefly ) warped product manifolds . If M is , then we give a condition on the warping function that M is a pseudosymmetric space and N is a space of constant curvature. If M is , then we show that (i) N is Ricci symmetric and (ii) M is if and only if the tensor T defined by (2.6) sat...

The object of the paper is to investigate almost alpha-cosymplectic (κ,μ,ν) spaces. Some results on almost alpha-cosymplectic (κ,μ,ν) spaces with certain conditions are obtained. Finally, we give an example on 3-dimensional case.

An indefinite Riemannian manifold is called neutral it its index is equal to one half of its dimension and an indefinite Kaehler manifold is called neutral Kaehler if its complex index is equal to the half of its complex dimension. In the first part of this article, we extend the notion of slant sur-faces in Lorentzian Kaehler surfaces to slant sub...

In this study we consider the focal curve Cγ of a space curve γ and its focal curvatures. We characterize some special types of ruled surface, choosing one of the base curves or director curves as the focal curve of the space curve γ. Finally we construct new types of ruled surface and calculate their distinguished parameters. We give necessary and...

In the present article we study the rotational embedded surfaces in 4 . The rotational embedded surface was first studied by G. Ganchev and V. Milousheva as a surface in 4 . The Otsuki (non-round) sphere in 4 is one of the special examples of this surface. Finally, we give necessary and sufficient conditions for the flat Ganchev-Milousheva rotation...

In the present paper we classify N(k)-contact metric manifolds which satisfy P(�,X) ·R = 0, R(�,X) · P = 0, P(�,X) · S = 0, P(�,X) · P = 0 and P(�,X) · Z = 0 where P is the Weyl projective curvature tensor and Z is the concircular curvature tensor.

In this study we consider the focal curve C γ of a space curve γ and its focal curvatures. We characterize some special types of ruled surface, choosing one of the base curves or director curves as the focal curve of the space curve γ. Finally we construct new types of ruled surface and calculate their distinguished parameters. We give necessary an...

The object of this paper is to study α-Kenmotsu manifolds which are certain from almost contact Riemannian manifolds satisfying some certain conditions. We first examine the generalized recurrent α-Kenmotsu manifolds, and next we give some relations about Ricci semi-symmetric and D-conformal curvature tensors. We show that Ricci semi-symmetric α-Ke...

In this paper we prove Chen inequalities for submanifolds of a locally conformal almost cosymplectic manifold N2m+1(c) of constant φ{symbol}-sectional curvature c endowed with a semi-symmetric metric connection, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and sectional curvatures, Ricci cu...

We consider φ-conformally flat, φ-conharmonically flat, φ-projectively flat and φ-concircularly flat Lorentzian α-Sasakian manifolds. In all cases, we get that the manifold will be an η-Einstein manifold.

In the present study, we consider isometric immersions \({f : M \rightarrow \tilde{M}(c)}\) of (2n + 1)-dimensional invariant submanifold M 2n+1 of (2m + 1) dimensional Sasakian space form \({\tilde{M}^{2m+1}}\) of constant \({ \varphi}\)-sectional curvature c. We have shown that if f satisfies the curvature condition \({\overset{\_}{R}(X, Y) \cdot...

Let M be an n-dimensional totally real minimal submanifold in CPn. We prove that if M is semi-parallel and the scalar curvature τ, , then M is an open part of the Clifford torus Tn ⊂ CPn. If M is semi-parallel and the scalar curvature τ, , then M is an open part of the real projective space RPn.

The object of the present paper is to study a type of Riemannian manifolds called generalized recurrent manifolds. We have constructed two concrete examples of such a manifold whose scalar curvature is non-zero non-constant. Some other properties have been considered. Among others it is shown that on a generalized recurrent manifold with constant s...

We study on semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds. We prove that a (2n + 1)-dimensional Sasakian hypersurface M of a (2n+2)-dimensional Kaehler manifold fM2n+2 is semiparallel if and only if it is totally umbilical with unit mean curvature, if dimM = 3 and fM4 is a Calabi-Yau manifold, then fM is at at each...

We consider semiparallel and 2-semiparallel invariant submanifolds of Lorentzian para-Sasakian manifolds. We show that these submanifolds are totally geodesic. We also consider invariant submanifolds of Lorentzian para-Sasakian manifolds satisfying the conditions Z(X, Y ) · α = 0 and Z(X, Y ) · ∇α = 0 with τ �= n(n − 1). Under these conditions, we...

We consider P-Sasakian manifolds satisfying the conditions R·P=0, P·R=0, C·P=0, P·C=0 and R·L=0, where R is the Riemannian curvature tensor, P is the Weyl conformal curvature tensor and L is the contact Ricci tensor.

The purpose of this paper is to study generalized − φ recurrent Kenmotsu

Let ˜ M, 2 (n + 2)c). 2000 Mathematics Subject Classification: Primary 53B20, 53B25, 53B50; Sec-

We study Riemannian manifolds M admitting a semi-symmetric metric connection ($) over tilde such that the vector field U is a parallel unit vector field with respect to the Levi-Civita connection del. We prove that ($) over tilde .R = 0 if and only if M is semisymmetric; if ($) over tilde .R = 0 or R.($) over tilde - ($) over tilde .R = 0 or M is s...

Let \(\widetilde{M}^{2n+1}(c)\) be (2n + 1)-dimensional Sasakian space form of constant ϕ-sectional curvature (c) and M n be an n -dimensional C-totally real, minimal submanifold of \(\widetilde{M}^{2n+1}(c)\). We prove that if M n is pseudo-parallel and \(Ln-\frac{1}{4}(n(c + 3) + c - 1)\ge 0\), then M n is totally geodesic.

We obtain some classification results and the stability conditions of nonminimal biharmonic anti-invariant submanifolds in Sasakian space forms.

In the present study, we considered 3-dimensional generalized (κ, μ)-contact metric manifolds. We proved that a 3-dimensional generalized (κ, μ)-contact metric manifold is not locally φ-symmetric in the sense of Takahashi. However such a manifold is locally φ-symmetric provided that κ and μ are constants. Also it is shown that if a 3-dimensional ge...

Recently, Chen established a general sharp inequality for warped products in real space forms. As applications, he obtained obstructions to minimal isometric immersions of warped products into real space forms. Afterwards, Matsumoto and one of the present authors proved the Sasakian version of this inequality. In the present paper, we obtain sharp...

We classify Lorentzian para-Sasakian manifolds which satisfy P ¢ C = 0, Z ¢ C = LCQ(g;C); P ¢ Z ¡ Z ¢ P = 0, and P ¢ Z + Z ¢ P = 0; where P is the v¡Weyl projective tensor, Z is the concircular tensor, and C is the Weyl conformal curvature tensor.

Recently, Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-war...

In this study, a geometric and experimental work of the urinary bladder of a dog is presented. Experimentally, the diameters on the neck (collum vesicae region), the body (corpus vesicae region), the bottom (apex region) and the longitudinal length of the urinary bladder were measured. Geometrically it was shown that the urinary bladder is comparab...

Biharmonic maps between Riemannian manifolds are defined as critical points of the bienergy and generalized harmonic maps. In this paper, we give necessary and sufficient conditions for nonharmonic Legendre curves and anti-invariant surfaces of 3-dimensional (κ,μ)-manifolds to be biharmonic.

Biharmonic maps between Riemannian manifolds are defined as critical points of the bienergy and generalized harmonic maps. In this paper, we give necessary and sufficient conditions for nonharmonic Legendre curves and anti-invariant surfaces of 3-dimensional (κ,μ)-manifolds to be biharmonic.

The present paper deals with Lorentzian α-Sasakian manifolds with confor-mally flat and quasi conformally flat curvature tensor. It is shown that in both cases, the manifold is locally isometric with a sphere S 2 n +1 (c). Further it is shown that an Lorentzian α-Sasakian manifold with R(X, Y).C = 0 is locally isometric with a sphere S 2 n +1 (c),...

The object of the present paper is to study a K-contact η-Einstein manifold satisfying a certain condition on the curvature tensor.

We consider hypersurfaces of a semi-Euclidean spaces satisfying some curvature condition of pseudosymmetry type related to solutions of the P.J. Ryan problem of the equivalence of semisymmetry and Ricci-semisymmetry on hypersurfaces.

We obtain certain inequalities involving several intrinsic invariants namely scalar curvature, Ricci curvature and $k$-Ricci curvature, and main extrinsic invariant namely squared mean curvature for submanifolds in a locally conformal almost cosymplectic manifold with pointwise constant $% \phi $-sectional curvature. Applying these inequalities we...

It is shown that every locally Euclidean 2-parallel submanifold of a space form has harmonic curvature vector (i.e., is weak biharmonic). In four-dimensional Euclidean space, in the class of surfaces with flat connection ∇, whose one family of curvature lines consists of geodesics, a surface is weak biharmonic if and only if it is 2-parallel. M.S.C...

In this paper the authors investigate hypersurfaces M of a semi-Euclidean space Esn+1, n ≥ 4, satisfying (αC + βR) · H = LkQ(g, Hk), k = 1,2,3. Using obtained results they show additional curvature properties of investigated hypersurfaces.

In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Similar problems for submanifolds in complex space forms were studied by Matsumoto et al. In this paper, we obtain sharp relationships between the Ricci curvature and the...

Chen (1999) established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimension. Matsumoto (to appear) dealt with similar problems for sub-manifolds in complex space forms. In this article we obtain sharp relationships between the Ricci curvature and the...

Chen (1999) established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimension. Matsumoto (to appear) dealt with similar problems for submanifolds in complex space forms. In this article we obtain sharp relationships between the Ricci curvature and the s...

In the paper we prove that under some additional curvature condition the relations R · R = 0 and R · S = 0 are equivalent for hypersurfaces of semi-Euclidean spaces. We present also examples of hypersurfaces having the curvature tensor expressed by the square, in the sense of the Kulkarni - Nomizu product, of the Ricci tensor.

In the present study we consider pseudo Ricci-symmetric manifolds in the sense of M. C. Chaki. We show that pseudo Ricci-symmetric manifolds satisfying divR=0 (resp. divC=0) are Einstein (resp. Ricci flat) manifolds.

We consider pseudosymmetric and pseudo Ricci symmetric manifolds in the sense of M. C. Chaki. The case M is assumed to be a contact metric manifold with belonging to (k, µ)-nullity distribution.

B.Y. Chen initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold (see (Ch3)). Inspired by Chen's denition, F. Decruyenaere, F. Dillen, L. Verstraelen and L. Vrancken (in (DDVV)) studied the tensor product of two immersions of, in general, dierent manifolds; un- der certain conditions, this realizes an i...