Conformal Semi-Invariant Submersions from Almost ...

Acta Math Vietnam DOI 10.1007/s40306-016-0193-9

Conformal Semi-Invariant Submersions from Almost Product Riemannian Manifolds Mehmet Akif Akyol1

Received: 13 January 2016 / Revised: 18 July 2016 / Accepted: 27 July 2016 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Abstract As a generalization of semi-invariant submersions, we introduce conformal semiinvariant submersions from almost product Riemannian manifolds onto Riemannian manifolds. We give examples, and investigate the geometry of foliations which arise from the definition of a conformal submersion and show that there are certain product structures on the total space of a conformal semi-invariant submersion. Moreover, we also find necessary and sufficient conditions for a conformal semi-invariant submersion to be totally geodesic. Keywords Almost product Riemannian manifold · Riemannian submersion · Semi-invariant submersion · Conformal submersion · Conformal semi-invariant submersion Mathematics Subject Classification (2010) Primary 53C42 · 53C43 · Secondary 53C15

1 Introduction One of the main methods to compare two manifolds and transfer certain structures from a manifold to another manifold is to define appropriate smooth maps between them. In particular, Riemannian submersions are fundamentally important in several areas of Riemannian geometry. For instance, it is a classical and important problem in Riemannian geometry to construct Riemannian manifolds with positive or non-negative sectional curvature. Riemannian submersions between Riemannian manifolds are important geometric structures. Riemannian submersions between Riemannian manifolds were studied by O’Neill [12] and

 Mehmet Akif Akyol

[email protected]; [email protected]

1

Faculty of Arts and Sciences, Department of Mathematics, Bing¨ol University, 12000, Bing¨ol, Turkey

M. A. Akyol

Gray [7]. In [16], the Riemannian submersions were considered between almost Hermitian manifolds by Watson under the name of almost Hermitian submersions. In this case, the Riemannian submersion is also an almost complex mapping and consequently the vertical and horizontal distribution are invariant with respect to the almost complex structure of the total manifold of the submersion. The study of anti-invariant Riemannian submersions from almost Hermitian manifolds were initiated by S¸ahin [14]. In this case, the fibres are antiinvariant with respect to the almost complex structure of the total manifold. In [15], he also defined and studied the notion of semi-invariant Riemannian submersions from almost Hermitian manifolds. In [9], G¨und¨uzalp investigated the notion of anti-invariant Riemannian submersions from almost product Riemannian manifolds and studied total manifolds. In particular, he investigated the foliations of total manifold and give necessary and sufficient conditions for total geodesicness. On the other hand, as a generalization of Riemannian submersions, horizontally conformal submersions are defined as follows [4]: Suppose that (M, gM ) and (B, gB ) are Riemannian manifolds and π : M −→ B is a smooth submersion. Then π is called a horizontally conformal submersion, if there is a positive function λ such that λ2 gM (X, Y ) = gB (π∗ X, π∗ Y ) for every X, Y ∈ ((kerπ∗ )⊥ ). It is obvious that every Riemannian submersion is a particular horizontally conformal submersion with λ = 1. We note that horizontally conformal submersions are special horizontally conformal maps which were introduced independently by Fuglede [5] and Ishihara [11]. We also recall that a horizontally conformal submersion π : M −→ B is said to be horizontally homothetic if the gradient of its dilation λ is vertical, i.e.,

H(gradλ) = 0

(1.1)

at p ∈ M, where H is the projection on the horizontal space (kerπ∗ )⊥ . One can see that Riemannian submersions are very special maps comparing with conformal submersions. Although contrary to isometries, conformal maps do not preserve distance between points, but they preserve angles between vector fields. This property enables one to transfer certain properties of a manifold to another manifold by deforming such properties. As a generalization of holomorphic submersions, conformal holomorphic submersions were studied by Gudmundsson and Wood [8]. They obtained necessary and sufficient conditions for conformal holomorphic submersions to be harmonic morphisms. Recently, we introduce conformal anti-invariant Riemannian submersions from almost product Riemannian manifolds onto Riemannianian manifolds and investigate the geometry of such submersions [1]. In this paper, as a generalization of conformal anti-invariant submersions, we define and study conformal semi-invariant submersions from almost product Riemannian manifolds onto Riemannian manifolds (see also [2] and [3]). The paper is organized as follows. In the second section, we gather main notions and formulas for other sections. In Section 3, we introduce conformal semi-invariant submersions from almost product Riemannian manifolds onto Riemannian manifolds, give examples and investigate the geometry of leaves of the horizontal distribution and the vertical distribution. In this section, we also show that there are certain product structures on the total space of a conformal semi-invariant submersion. In Section 4, we find necessary and sufficient conditions for a conformal semi-invariant submersion to be totally geodesic.

Conformal Semi-˙Invariant Submersions...

2 Preliminaries In this section, we define almost product Riemannian manifolds, recall the notion of (horizontally) conformal submersions between Riemannian manifolds, and give a brief review of basic facts of (horizontally) conformal submersions. Let M be a m-dimensional manifold with a tensor F of a type (1,1) such that F 2 = I, (F = I ). Then, we say that M is an almost product manifold with almost product structure F. We put 1 1 P = (I + F ), Q = (I − F ). 2 2 Then, we get P + Q = I, P 2 = P , Q2 = Q, P Q = QP = 0, F = P − Q. Thus, P and Q define two complementary distributions P and Q. We easily see that the eigenvalues of F are +1 and −1. If an almost product manifold M admits a Riemannian metric g such that g(F X, F Y ) = g(X, Y ) (2.1) for any vector fields X and Y on M, then M is called an almost product Riemannian manifold, denoted by (M, g, F ). Denote the Levi-Civita connection on M with respect to g by M ∇ . Then, M is called a locally product Riemannian manifold [19] if F is parallel with M respect to ∇ , i.e., M (2.2) ∇X F = 0, X ∈ (T M). Conformal submersions belong to a wide class of conformal maps that we are going to recall their definition, but we will not study such maps in this paper. Definition 2.1 ([4]) Let ϕ : (M m , g) −→ (N n , h) be a smooth map between Riemannian manifolds, and let x ∈ M. Then, ϕ is called horizontally weakly conformal or semi conformal at x if either (i) (ii)

dϕx = 0, or dϕx maps the horizontal space Hx = (ker(dϕx ))⊥ conformally onto Tϕ(x) N , i.e., dϕx is surjective and there exists a number (x) = 0 such that h(dϕx X, dϕx Y ) = (x)g(X, Y ) (X, Y ∈ Hx ).

(2.3)

A point x of type (i) in Definition 2.1 is called a critical point of ϕ; we shall call a point of type (ii) a regular point. At a critical point, dϕx has rank 0; at a regular point, dϕx has rank n and ϕ is a submersion. The number (x) is called √ the square dilation (of ϕ at x); it is necessarily non-negative; its square root λ(x) = (x) is called the dilation (of ϕ at x). The map ϕ is called horizontally weakly conformal or semi conformal (on M) if it is horizontally weakly conformal at every point of M. If ϕ has no critical points, then we call it a (horizontally) conformal submersion. Next, we recall the following definition from [4]. Let F : M −→ N be a submersion. A vector field E on M is said to be projectable if there exists a vector field Eˇ on N , such that F∗ (Ex ) = Eˇ F (x) for all x ∈ M. In this case, E and Eˇ are called F − related. A horizontal vector field Y on (M, g) is called basic, if it is projectable. It is a well known fact, that Zˇ is a vector field on N , then there exists a unique basic vector field Z on M, such that Z and Zˇ ˇ are F − related. The vector field Z is called the horizontal lift of Z.

M. A. Akyol

The fundamental tensors of a submersion were introduced in [12]. They play a similar role to that of the second fundamental form of an immersion. More precisely, O’Neill’s tensors T and A are defined for vector fields E, F on M by M

M

M

M

AE F = V ∇HE HF + H∇HE V F, TE F = H∇V E V F + V ∇V E HF,

(2.4)

where V and H are the vertical and horizontal projections (see [6]). On the other hand, from (2.4), we have M ∇V W = TV W + ∇ˆ V W M

M

∇ V X = H ∇V X + T V X M

M

∇ X V = A X V + V ∇X V M

M

∇X Y = H ∇ X Y + A X Y

(2.5) (2.6) (2.7) (2.8)

M for X, Y ∈ ((kerF∗ )⊥ ) and V , W ∈ (kerF∗ ), where ∇ˆ V W = V ∇V W . If X is basic, M then H∇V X = AX V . It is easily seen that for x ∈ M, X ∈ Hx and V ∈ Vx the linear operators TV , AX : Tx M −→ Tx M are skew-symmetric. We also see that the restriction of T to the vertical distribution T |V ×V is exactly the second fundamental form of the fibres of F . Since TV is skew-symmetric, we get: F has totally geodesic fibres if and only if T ≡ 0. We now recall the notion of harmonic maps between Riemannian manifolds. Let (M, gM ) and (N, gN ) be Riemannian manifolds and suppose that ϕ : M −→ N is a smooth map between them. Then the differential of ϕ∗ of ϕ can be viewed as a section of the bundle H om(T M, ϕ −1 T N ) −→ M, where ϕ −1 T N is the pullback bundle which has fibres (ϕ −1 T N )p =Tϕ(p) N , p ∈ M. H om(T M, ϕ −1 T N ) has a connection ∇ induced from the Levi-Civita connection ∇ M and the pullback connection. Then the second fundamental form of ϕ is given by

ϕ

M Y) (∇ϕ∗ )(X, Y ) = ∇X ϕ∗ (Y ) − ϕ∗ (∇X

(2.9)

for X, Y ∈ (T M), where ∇ ϕ is the pullback connection. It is known that the second fundamental form is symmetric. Lemma 2.1 [18] Let (M, gM ) and (N, gN ) be Riemannian manifolds and suppose that ϕ : M −→ N is a smooth map between them. Then we have ϕ

ϕ

∇X ϕ∗ (Y ) − ∇Y ϕ∗ (X) − ϕ∗ ([X, Y ]) = 0

(2.10)

for X, Y ∈ (T M). Finally, we recall the following lemma from [4]. Lemma 2.2 Suppose that π : M −→ N is a horizontally conformal submersion. Then, for any horizontal vector fields X, Y and vertical fields V , W , we have (i) (∇π∗ )(X, Y ) = X(ln λ)π∗ (Y ) + Y (ln λ)π∗ (X) − g(X, Y )π∗ (grad ln λ); (ii) (∇π∗ )(V , W ) = −π∗ (TV W ); M V ) = −π (A V ). (iii) (∇π∗ )(X, V ) = −π∗ (∇X ∗ X

Conformal Semi-˙Invariant Submersions...

˙ 3 Conformal Semi-Invariant Submersions In this section, we define conformal semi-invariant submersions from an almost product Riemannian manifold onto a Riemannian manifold, and we investigate the integrability of distributions and show that there are certain product structures on the total space of such submersions. Definition 3.1 Let M be an almost product Riemannian manifold with Riemannian metric gM and almost product structure F and B be a Riemannian manifold with Riemannian metric gB . A horizontally conformal submersion π : M −→ B with dilation λ is called a conformal semi-invariant submersion if there is a distribution D1 ⊆ kerπ∗ such that kerπ∗ = D1 ⊕ D2

(3.1)

F (D1 ) = D1 , F (D2 ) ⊆ (kerπ∗ )⊥ ,

(3.2)

and where D2 is orthogonal complementary to D1 in kerπ∗ . We now give some examples of conformal semi-invariant submersions. Example 3.1 Every anti-invariant submersion from an almost product Riemannian manifold onto a Riemannian manifold is a conformal semi-invariant submersion with λ = I and D1 = {0}, where I denotes the identity function [9]. Example 3.2 Every invariant submersion from an almost product Riemannian manifold onto a Riemannian manifold is a conformal semi-invariant submersion with λ = I and D2 = {0} [10]. Example 3.3 Every semi-invariant submersion from an almost product Riemannian manifold to a Riemannian manifold is a conformal semi-invariant submersion with λ = I. We say that a conformal semi-invariant submersion is proper if λ  = I . We now present an example of a proper conformal semi-invariant submersion. Note that given an Euclidean space R 5 with coordinates (x1 , . . . , x5 ), we can canonically choose an almost product structure F on R 5 as follows: F : R 5 −→ R 5 , F (x1 , x2 , x3 , x4 , x5 ) = (x2 , x1 , x3 , x5 , x4 ). Example 3.4 Let π be a submersion defined by π:

−→ R2 R5 (x1 , x2 , x3 , x4 , x5 ) (sin x2 cosh x3 , cos x2 sinh x3 ).

Then it follows that kerπ∗ = span{V1 = ∂x1 , V2 = ∂x4 , V3 = ∂x5 } and (kerπ∗ )⊥ = span{H1 = cos x2 cosh x3 ∂x2 + sin x2 sinh x3 ∂x3 , H2 = − sin x2 sinh x3 ∂x2 + cos x2 cosh x3 ∂x3 }.

M. A. Akyol

Hence, we have F V2 = V3 , cos x2 cosh x3 sin x2 sinh x3 F V1 = H1 − H2 . 2 2 2 2 2 cos x2 cosh x3 + sin x2 sinh x3 cos x2 cosh2 x3 + sin2 x2 sinh2 x3 Thus, it follows that D1 = span{V2 , V3 } and D2 = span{V1 }. Also by direct computations, we get π∗ H1 = (cos2 x2 cosh2 x3 + sin2 x2 sinh2 x3 )∂y1 and π∗ H2 = (cos2 x2 cosh2 x3 + sin2 x2 sinh2 x3 )∂y2 . Hence, we have g2 (π∗ H1 , π∗ H1 ) = (cos2 x2 cosh2 x3 + sin2 x2 sinh2 x3 )g1 (H1 , H1 ) and g2 (π∗ H2 , π∗ H2 ) = (cos2 x2 cosh2 x3 + sin2 x2 sinh2 x3 )g1 (H2 , H2 ), where g1 and g2 denote the standard metrics (inner products) of R 5 and R 2 . Thus π is a conformal semi-invariant submersion with λ2 = cos2 x2 cosh2 x3 + sin2 x2 sinh2 x3 . Let π be a conformal semi-invariant submersion from an almost product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). We denote the complementary distribution to F D2 in (kerπ∗ )⊥ by μ. For V ∈ (kerπ∗ ), we write F V = φV + ωV

(3.3)

where φV ∈ (D1 ) and ωV ∈ (F D2 ). Also for X ∈ ((kerπ∗

)⊥ ),

we have

F X = B X + C X,

(3.4)

where B X ∈ (D2 ) and C X ∈ (μ). Then by using (3.3), (3.4), (2.5) and (2.6), we get M

(∇V φ)W = B TV W − TV ωW

(3.5)

M

(∇V ω)W = C TV W − TV φW

(3.6)

for V , W ∈ (kerπ∗ ), where (∇V φ)W = ∇ˆ V φW − φ ∇ˆ V W M

and

(∇V ω)W = H∇V ωW − ω∇ˆ V W. We now investigate the integrability of the distributions D1 and D2 . M

M

Theorem 3.1 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). Then the distribution D1 is integrable if and only if (∇π∗ )(V1 , F U1 ) − (∇π∗ )(U1 , F V1 ) ∈ (π∗ (μ))

(3.7)

for U1 , V1 ∈ (D1 ) and Z ∈ (D2 ). Proof We note that the distribution D1 is integrable if and only if gM ([U1 , V1 ], Z) = gM ([U1 , V1 ], W ) = 0 for U1 , V1 ∈ (D1 ), Z ∈ (D2 ) and W ∈ ((ker π∗ )⊥ ). Since ker π∗ is integrable gM ([U1 , V1 ], W ) = 0. Thus, D1 is integrable if and only if gM ([U1 , V1 ], Z) = 0. Moreover, by using (2.1), (2.2), (2.5) and (3.2), we have M

M

gM ([U1 , V1 ], Z) = g1 (H∇U1 F V1 , F Z) − gM (H∇V11 F U1 , F Z).

Conformal Semi-˙Invariant Submersions...

Using Lemma 2.2 and taking into account that π is a conformal submersion, we obtain gM ([U1 , V1 ], Z) =

1 1 M M gB (π∗ (∇U1 F V1 ), π∗ F Z) − 2 gB (π∗ (∇V1 F U1 ), π∗ F Z). λ2 λ

Then using (2.9) we get 1 gB ((∇π∗ )(V1 , F U1 ) − (∇π∗ )(U1 , F V1 ), π∗ F Z). λ2 The distribution D1 is integrable if and only if (3.7) is satisfied. gM ([U1 , V1 ], Z) =

In a similar way, we get: Theorem 3.2 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). Then the distribution D2 is integrable if and only if gM (TU2 F V2 , Z) = gM (TV2 F U2 , Z)

(3.8)

for U2 , V2 ∈ (D2 ) and Z ∈ (D1 ). Proof We note that the distribution D2 is integrable if and only if gM ([U2 , V2 ], F Z) = gM ([U2 , V2 ], W ) = 0 for U2 , V2 ∈ (D2 ), Z ∈ (D1 ) and W ∈ ((ker π∗ )⊥ ). Since ker π∗ is integrable gM ([U2 , V2 ], W ) = 0. Thus D2 is integrable if and only if gM ([U2 , V2 ], F Z) = 0. Moreover, by using (2.1) and (2.2), we have M

M

gM ([U2 , V2 ], F Z) = gM (∇U2 F V2 , Z) − gM (∇V2 F U2 , Z). Using (2.5) and (2.6), we get M

M

gM ([U2 , V2 ], F Z) = gM (TU2 F V2 + H∇U2 F V2 , Z) − gM (TV2 F U2 + H∇V2 F U2 , Z) = gM (TU2 F V2 , Z) − gM (TV2 F U2 , Z). From the above equation, the distribution D2 is integrable if and only if (3.8) is satisfied. Since (kerπ∗ )⊥ = F (D2 ) ⊕ μ and π is a conformal semi-invariant submersion from an almost product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ), for X ∈ (D2 ) and Y ∈ (μ), we have λ12 gB (π∗ F X, π∗ Y ) = gM (F X, Y ) = 0. This implies that the distributions π∗ (F D2 ) and π∗ (μ) are orthogonal. Now, we investigate the geometry of the leaves of the distributions D1 and D2 . Theorem 3.3 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ). Then D1 defines a totally geodesic foliation on M if and only if (∇π∗ )(F V1 , U1 ) ∈ (π∗ μ) and 1 gB ((∇π∗ )(U1 , F V1 ), π∗ C Z) = gM (∇ˆ U1 F V1 , B Z) λ2 for U1 , V1 ∈ (D1 ) and Z ∈ ((kerπ∗ )⊥ ). Proof The distribution D1 defines a totally geodesic foliation on M if and only if M M gM (∇U1 V1 , U2 ) = 0 and gM (∇U1 V1 , Z) = 0 for U1 , V1 ∈ (D1 ), U2 ∈ (D2 ) and

M. A. Akyol

Z ∈ ((kerπ∗ )⊥ ). Since M is a locally product Riemannian manifold, using (2.1) and (2.2), we get M M gM (∇U1 V1 , U2 ) = gM (H∇U1 F V1 , F U2 ). Since π is a conformal semi-invariant submersion, using (2.9), we have M

gM (∇U1 V1 , U2 ) = − λ12 gB ((∇π∗ )(F V1 , U1 ), π∗ F U2 ).

(3.9)

On the other hand, by using (2.1), (2.2), (2.5) and (3.4), we derive M

M

M

gM (∇U1 V1 , Z) = gM (∇U1 F V1 , B Z) + gM (∇U1 F V1 , C Z). Now, from (2.6) and (2.9), we have 1 M gM (∇U1 V1 , Z) = gM (∇ˆ U1 F V1 , B Z) − 2 gB ((∇π∗ )(U1 , F V1 ), π∗ C Z). λ Thus, the proof follows from (3.9) and (3.10).

(3.10)

For D2 , we have the following result. Theorem 3.4 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ). Then D2 defines a totally geodesic foliation on M if and only if (∇π∗ )(U2 , F U1 ) ∈ (π∗ μ) and 1 − 2 gB (∇Fπ V2 π∗ F U2 , π∗ F C Z) = gM (TU2 F V2 , B Z)+gM (U2 , V2 )gM (Hgrad ln λ, F C Z) λ for U2 , V2 ∈ (D2 ), U1 ∈ (D1 ) and Z ∈ ((kerπ∗ )⊥ ). Proof The distribution D2 defines a totally geodesic foliation on M if and only if M M gM (∇U2 V2 , U1 ) = 0 and gM (∇U2 V2 , Z) = 0 for U2 , V2 ∈ (D2 ), U1 ∈ (D1 ) and Z ∈ ((kerπ∗ )⊥ ). Since π is a conformal semi-invariant submersion, using (2.1), (2.2) and (2.9), we get 1 gB ((∇π∗ )(U2 , F U1 ), π∗ F V2 ). λ2 On the other hand, by using (2.1), (2.2), (2.5) and (3.4), we derive M

gM (∇U2 V2 , U1 ) =

M

(3.11)

M

gM (∇U2 V2 , Z) = −gM (F V2 , TU2 B Z) + gM ([U2 , F V2 ] + ∇F V2 U2 , C Z). Since [U2 , F V2 ] ∈ (kerπ∗ ), we obtain M

M

gM (∇U2 V2 , Z) = gM (TU2 F V2 , B Z) + gM (∇F V2 F U2 , F C Z). Now using (2.9), we obtain M

gM (∇U2 V2 , Z) = gM (TU2 F V2 , B Z) + gM (V2 , U2 )gM (Hgrad ln λ, F C Z) 1 gB (∇Fπ V2 π∗ F U2 , π∗ F C Z). λ2 Thus, the proof follows from (3.11) and (3.12). +

From Theorem 3.3 and Theorem 3.4, we have the following theorem.

(3.12)

Conformal Semi-˙Invariant Submersions...

Theorem 3.5 Let π : (M, gM , F ) −→ (B, gB ) be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). Then the fibers of π are locally product manifold if and only if (∇π∗ )(F V1 , U1 ) ∈ (π∗ μ) and (∇π∗ )(F U1 , U2 ) ∈ (π∗ μ) for any U1 , V1 ∈ (D1 ), U2 ∈ (D2 ). We now study the integrability of the distribution (kerπ∗ )⊥ and then we investigate the geometry of leaves of kerπ∗ and (kerπ∗ )⊥ . Theorem 3.6 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). Then the distribution (kerπ )⊥ is integrable if and only if / (D1 ) F (AZ C W − AW C Z) + AZ ωB W − AW ωB Z ∈ and 1 π gB (∇W π∗ C Z− ∇Zπ π∗ C W, π∗ F U2 ) = gM (AW B Z−AZ B W− C W (ln λ)Z+ C Z(ln λ)W λ2 +2gM (Z, C W )grad ln λ, F U2 ) for Z, W ∈ ((kerπ∗ )⊥ ) and W ∈ (D2 ). Proof The distribution (kerπ∗ )⊥ is integrable on M if and only if gM ([Z, W ], U1 ) = 0 and gM ([Z, W ], U2 ) = 0 for Z, W ∈ ((kerπ∗ )⊥ ), U1 ∈ (D1 ) and U2 ∈ (D2 ). Since M is a locally product Riemannian manifold, using (2.1), (2.2) and (3.4), we get M

M

M

M

M

gM ([Z, W ], U1 ) = gM (∇Z B W, F U1 ) + gM (∇Z C W, F U1 ) − gM (∇W B Z, F U1 ) M

−gM (∇W C Z, F U1 ). Also using (2.8), we get M

gM ([Z, W ], U1 ) = gM (∇Z F B W, U1 ) + gM (∇Z C W, F U1 ) − gM (∇W F B Z, U1 ) M

−gM (∇W C Z, F U1 ). From (2.7) and (3.3), we derive gM ([Z, W ], U1 ) = gM (AZ ωB W − AW ωB Z + F AZ C W − F AW C Z, U1 ).

(3.13)

On the other hand, from (2.1), (2.2) and (3.4), we derive M

M

M

gM ([Z, W ], U2 ) = gM (∇Z B W, F U2 ) + gM (∇Z C W, F U2 ) − gM (∇W B Z, F U2 ) M

−gM (∇W C Z, F U2 ).

M. A. Akyol

Since π is a conformal submersion, using (2.9) and Lemma 2.2, we arrive at gM ([Z, W ], U2 ) =

1 1 M M gB (π∗ (∇Z B W ), π∗ F U2 ) − 2 gB (π∗ (∇W B Z), π∗ F U2 ) λ2 λ 1 + 2 gB {−Z(ln λ)π∗ C W − C W (ln λ)π∗ Z+ gM (Z, C W )π∗ (grad ln λ) λ +∇Zπ π∗ C W, π∗ F U2 } 1 − 2 gB {−W (ln λ)π∗ C Z− C Z(ln λ)π∗ W + gM (W, C Z)π∗ (grad ln λ) λ π +∇W π∗ C Z, π∗ F U2 }.

Thus from (2.9) and (2.7), we have gM ([Z, W ], U2 ) = 1 gM (grad ln λ, Z)gB (π∗ C W, π∗ F U2 ) λ2 1 1 − 2 gM (grad ln λ, C W )gB (π∗ Z, π∗ F U2 ) + 2 gM (Z, C W )gB (π∗ (grad ln λ), π∗ F U2 ) λ λ 1 1 π + 2 gB (∇Z π∗ C W, π∗ F U2 ) + 2 gM (grad ln λ, W )gB (π∗ C Z, π∗ F U2 ) λ λ 1 1 + 2 gM (grad ln λ, C Z)gB (π∗ W, π∗ F U2 ) − 2 gM (W, C Z)gB (π∗ (grad ln λ), π∗ F U2 ) λ λ 1 π π∗ C Z, π∗ F U2 ). − 2 gB (∇W λ Moreover, using Definition 3.1, we obtain gM (AZ B W, F U2 ) − gM (AW B Z, F U2 ) −

gM ([Z, W ], U2 ) = gM (AZ B W −AW B Z − C W (ln λ)Z + C Z(ln λ)W 1 π π∗ C Z−∇Zπ π∗ C W, π∗ F U2 ). +2gM (Z, C W )grad ln λ, F U2 )− 2 gB(∇W λ (3.14) Thus the proof follows from (3.13) and (3.14). The next theorem gives a necessary and sufficient condition for a conformal submersion to be a homothetic map. Theorem 3.7 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ) with integrable distribution (kerπ∗ )⊥ . Then π is a horizontally homothetic map if and only if the following condition π λ2 gM (AZ B W − AW B Z, F U2 ) = gB (∇W π∗ C Z − ∇Zπ π∗ C W, π∗ F U2 )

(3.15)

is satisfied for Z, W ∈ ((kerπ∗ )⊥ ) and U2 ∈ (D2 ).

Proof From (3.14), we have gM ([Z, W ], U2 )= gM (AZ B W − AW B Z − C W (ln λ)Z + C Z(ln λ)W 1 π π∗ C Z−∇Zπ π∗ C W, π∗ F U2 ) +2gM (Z, C W )grad ln λ, F U2 )− 2 gB (∇W λ

Conformal Semi-˙Invariant Submersions...

for Z, W ∈ ((kerπ∗ )⊥ ) and U2 ∈ (D2 ). If π is a horizontally homothetic map, then we get (3.15). Conversely, if (3.15) is satisfied, then we get gM (−gM (grad ln λ, C W )Z + gM (grad ln λ, C Z)W + 2gM (Z, C W )grad ln λ, F U2 ) = 0. (3.16) Now, taking W = F U2 for U2 ∈ (D2 ) in (3.16), we have gM (grad ln λ, C Z)gM (F U2 , F U2 ) = 0. Thus, λ is a constant on (μ). On the other hand, taking W = C Z for Z ∈ (μ) in (3.16) we obtain 2gM (Z, C 2 Z)gM (grad ln λ, F U2 ) = 2gM (Z, Z)gM (grad ln λ, F U2 ) = 0. From the above equation, λ is a constant on (F D2 ). This completes the proof. As conformal version of anti-holomorphic semi-invariant submersion ([17]), a conformal semi-invariant submersion is called a conformal anti-holomorphic semi-invariant submersion if F (D2 ) = (kerπ∗ )⊥ . For a conformal anti-holomorphic semi-invariant submersion, we have the following. Corollary 3.1 Let π be a conformal anti-holomorphic semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). Then the following assertions are equivalent to each other: (i) (ii)

(kerπ∗ )⊥ is integrable; gB (π∗ F U1 , (∇π∗ )(V , F U2 )) = gB (π∗ F V2 , (∇π∗ )(V , F U1 )) for U1 , U2 ∈ (D2 ) and V ∈ (kerπ∗ ).

Proof Since M is a locally product Riemannian manifold, using (2.1) and (2.2) we have M

M

gM ([F U1 , F U2 ], V ) = −gM (F U2 , ∇V F U1 ) + gM (F U1 , ∇V F U2 ) for U1 , U2 ∈ (D2 ) and V ∈ (kerπ∗ ). Since π is a conformal submersion, by using Lemma 2.2, we get 1 gM ([F U1 , F U2 ], V ) = 2 {gB (π∗ F U2 , (∇π∗ )(V , F U1 )) − gB (π∗ F U1 , (∇π∗ )(V , F U2 ))}. λ Thus, the proof is complete. For the geometry of leaves of the horizontal distribution, we have the following theorem. Theorem 3.8 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). Then the distribution (kerπ∗ )⊥ defines a totally geodesic foliation on M if and only if M

AZ C W + V ∇Z B W ∈ (D2 ) and λ2 {gM (AZ B W − C W (ln λ)Z + gM (Z, C W )grad ln λ, F U2 )} = gB (∇Zπ π∗ F U2 , π∗ C W ) for Z, W ∈ ((kerπ∗ )⊥ ), U1 ∈ (D1 ) and U2 ∈ (D2 ). Proof The distribution (kerπ∗ )⊥ defines a totally geodesic foliation on M if and only if M

M

gM (∇Z W, U1 ) = 0 and gM (∇Z W, U2 ) = 0

M. A. Akyol

for Z, W ∈ ((kerπ∗ )⊥ ), U1 ∈ (D1 ) and U2 ∈ (D2 ). Then by using (2.1), (2.2), (2.7), (2.8) and (3.3), we get M

M

gM (∇Z W, U1 ) = gM (φ(V ∇Z B W + AZ C W ), U1 ).

(3.17)

On the other hand, from (2.1), (2.2) and (3.3), we get M

M

M

gM (∇Z W, U2 ) = −gM (B W, ∇Z F U2 ) − gM (C W, ∇Z F U2 ). Since π is a conformal submersion, using (2.8), (2.9) and Lemma 2.2, we arrive at 1 gM (grad ln λ, F U2 )gB (π∗ Z, π∗ C W ) λ2 1 1 − 2 gM (Z, F U2 )gB (π∗ (grad ln λ), π∗ C W )− 2 gB (∇Zπ π∗ F U2 , π∗ C W ). λ λ The conformal semi-invariance of π implies that M

gM (∇Z W, U2 )=−gM (B W, AZ F U2 ) +

M

gM (∇Z W, U2 ) = gM (AZ B W − C W (ln λ)Z + gM (Z, C W )grad ln λ, F U2 ) 1 − 2 gB (∇Zπ π∗ F U2 , π∗ C W ). (3.18) λ Thus the proof follows from (3.17) and (3.18). Next, we give new conditions for conformal semi-invariant submersions to be horizontally homothetic maps. But we first give the following definition. Definition 3.2 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (B, gB ). Then we say that M D2 is parallel along (kerπ∗ )⊥ if ∇Z U2 ∈ (D2 ) for Z ∈ ((kerπ∗ )⊥ ) and U2 ∈ (D2 ). Corollary 3.2 Let π : (M, gM , F ) −→ (B, gB ) be a conformal semi-invariant submersion such that D2 is parallel along (kerπ∗ )⊥ . Then π is a horizontally homothetic map if and only if (3.19) λ2 gM (AZ B W, F U2 ) = gB (∇Zπ π∗ F U2 , π∗ C W )

for Z, W ∈ ((kerπ∗ )⊥ ) and U2 ∈ (D2 ). Proof (3.19) implies that

− gM (grad ln λ, C W )gM (Z, F U2 ) + gM (Z, C W )gM (grad ln λ, F U2 ) = 0.

(3.20)

Now, taking Z = F U2 for U2 ∈ (D2 ) in (3.20), we get gM (grad ln λ, C W )gM (F U2 , F U2 ) = 0. Thus, λ is a constant on (μ). On the other hand, taking Z = CW for W ∈ (μ) in (3.20) we derive gM (C W, C W )gM (grad ln λ, F U2 ) = 0. From the above equation, λ is a constant on (F D2 ). The converse is clear from (3.18). In particular, if π is a conformal anti-holomorphic semi-invariant submersion, then we have the following. Corollary 3.3 Let π be a conformal anti-holomorphic semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ). Then the following assertions are equivalent to each other:

Conformal Semi-˙Invariant Submersions...

(i) (ii)

(kerπ∗ )⊥ defines a totally geodesic foliation on M. (∇π∗ )(V , F W1 ) ∈ (π∗ (μ))

for W1 ∈ (D2 ) and V ∈ (kerπ∗ ). Next, we investigate the geometry of leaves of the distribution kerπ∗ . Theorem 3.9 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ). Then the distribution (kerπ∗ ) defines a totally geodesic foliation on M if and only if π λ2 {gM (C TU φV + AωV φU + gM (ωV , ωU )grad ln λ, Z)} = −gB (∇ωV π∗ Z, π∗ ωU )

and

TV ωU + ∇ˆ V φU ∈ (D1 ) for U, V ∈ (kerπ∗ ), Z ∈ (μ) and W ∈ (D2 ).

Proof The distribution (kerπ∗ ) defines a totally geodesic foliation on M if and only if M M gM (∇U V , Z) = 0 and gM (∇U V , F W ) = 0 for U, V ∈ (kerπ∗ ), Z ∈ (μ) and W ∈ (D2 ). Using (2.1), (2.2) and (3.3), we have M

M

M

M

gM (∇U V , Z) = gM (∇U φV , F Z) − gM (φU, ∇ωV Z) − gM (ωU, ∇ωV Z). Using (2.5), (2.8), (2.9), Lemma 2.2, and taking into account π is a conformal submersion, we arrive at 1 M gM(∇U V , Z)=gM(TU φV , F Z)−gM (φU,AωV Z)+ 2 gM (grad ln λ,Z)gB(π∗ ωV , π∗ ωU ) λ 1 π + 2 gB (∇ωV π∗ Z, π∗ ωU ). λ Hence, we obtain M

gM (∇U V , Z) = gM (C TU φV + AωV φU + gM (ωV , ωU )grad ln λ, Z) 1 π π∗ Z, π∗ ωU ). + 2 gB (∇ωV λ On the other hand, by using (2.1), (2.2) and (3.3), we get gM (∇U V , F W ) = gM (ω(∇ˆ V φU + TV ωU ), F W ). M

(3.21)

(3.22)

Thus, the proof follows from (3.21) and (3.22). Next, we give certain conditions for dilation λ to be constant on μ. We first give the following definition. Definition 3.3 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ). Then, we say that μ M is parallel along kerπ∗ if ∇U Z ∈ (μ) for Z ∈ (μ) and U ∈ (kerπ∗ ). Corollary 3.4 Let π : (M, gM , F ) −→ (B, gB ) be a conformal semi-invariant submersion such that μ is parallel along (kerπ∗ ). Then π is a constant on μ if and only if π λ2 gM (C TU φV + AωV φU, Z) = −gB (∇ωV π∗ Z, π∗ ωU )

for U, V ∈ (kerπ∗ ) and Z ∈ (μ).

(3.23)

M. A. Akyol

Proof If we have (3.23) then gM (ωV , ωU )gM (grad ln λ, Z) = 0 for Z ∈ ((kerπ∗ comes from (3.21).

)⊥ ).

(3.24)

From the above equation, λ is a constant on (μ). The converse

From Theorem 3.8 and Theorem 3.9, we have the following decomposition for total space. Theorem 3.10 Let π : (M, gM , F ) −→ (B, gB ) be a conformal semi-invariant submersion, where (M, gM , F ) is a locally product Riemannian manifold and (B, gB ) is a Riemannian manifold. Then M is a locally product manifold of the form M(kerπ∗ ) ×λ M(kerπ∗ )⊥ if and only if π λ2 {gM (C TU φV + AωV φU + gM (ωV , ωU )grad ln λ, Z)} = −gB (∇ωV π∗ Z, π∗ ωU ), TV ωU + ∇ˆ V φU ∈ (D1 )

and M

AZ C W + V ∇Z B W ∈ (D2 ) λ2 {gM (AZ B W − C W (ln λ)Z + gM (Z, C W )grad ln λ, F U2 )} = gB (∇Zπ π∗ F U2 , π∗ C W ) for Z, W ∈ ((kerπ∗ )⊥ ) and U, V ∈ (kerπ∗ ).

4 Total Geodesicness of the Conformal Semi-Invariant Submersions In this section, we obtain necessary and sufficient conditions for a conformal semi-invariant submersion to be totally geodesic. We recall that a differentiable map π between two Riemannian manifolds is called totally geodesic if (∇π∗ )(Z, W ) = 0 ∀Z, W ∈ (T M). A geometric interpretation of a totally geodesic map is that it maps every geodesic in the total manifold into a geodesic in the base manifold in proportion to arc lengths. We now present the following definition. Definition 4.1 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ). Then π is called a (F D2 , μ)-totally geodesic map if (∇π∗ )(F U, Z) = 0 for U ∈ (D2 ) and Z ∈ (μ). In the sequel, we show that this notion has an important effect on the geometry of the conformal submersion. Theorem 4.1 Let π be a conformal semi-invariant submersion from a locally product Riemannian manifold (M, gM , F ) to a Riemannian manifold (B, gB ). Then π is a (F D2 , μ)-totally geodesic map if and only if π is a horizontally homothetic map. Proof For U ∈ (D2 ) and Z ∈ (μ), from Lemma 2.2, we have (∇π∗ )(F U, Z) = F U (ln λ)π∗ Z + Z(ln λ)π∗ F U − gM (F U, Z)π∗ (grad ln λ).

Conformal Semi-˙Invariant Submersions...

From the above equation, if π is a horizontally homothetic map then (∇π∗ )(F U, Z) = 0. Conversely, if (∇π∗ )(F U, Z) = 0, we obtain F U (ln λ)π∗ Z + Z(ln λ)π∗ F U = 0.

(4.1)

Taking inner product in (4.1) with π∗ F U and since π is a conformal submersion, we have gM (grad ln λ, F U )gB (π∗ Z, π∗ F U ) + gM (grad ln λ, Z)gB (π∗ F U, π∗ F U ) = 0. The above equation implies that λ is a constant on (μ). On the other hand, taking inner product in (4.1) with π∗ Z, we have gM (grad ln λ, F U )gB (π∗ Z, π∗ Z) + gM (grad ln λ, Z)gB (π∗ F U, π∗ Z) = 0. From the above equation, it follows that λ is a constant on (F D2 ). Thus, λ is a constant on ((kerπ∗ )⊥ ). Hence the proof is complete. Finally, we give necessary and sufficient conditions for a conformal semi-invariant submersion to be totally geodesic. Theorem 4.2 Let π : (M, gM , F ) −→ (B, gB ) be a conformal semi-invariant submersion, where (M, gM , F ) is a locally product Riemannian manifold and (B, gB ) is a Riemannian manifold. Then π is a totally geodesic map if and only if (a) (b) (c) (d)

π is a horizontally homothetic map. C TU F V + ω∇ˆ U F V = 0 U, V ∈ (D1 ) M CH∇U F W + ωTU F W = 0 U ∈ (kerπ∗ ), W ∈ (D2 ) M TU B X + H∇U C X ∈ (F D2 ) and ∇ˆ U B X + TU C X ∈ (D1 ), U ∈ (kerπ∗ ), X ∈ ((kerπ∗ )⊥ ).

Proof (a) For any Z, W ∈ (μ), from Lemma 2.2, we derive (∇π∗ )(Z, W ) = Z(ln λ)π∗ W + W (ln λ)π∗ Z − gM (Z, W )π∗ (grad ln λ). It is obvious that if π is a horizontally homothetic map, then (∇π∗ )(Z, W ) = 0. Conversely, if (∇π∗ )(Z, W ) = 0, taking W = F Z in the above equation, we get Z(ln λ)π∗ F Z + F Z(ln λ)π∗ Z − gM (Z, F Z)π∗ (grad ln λ) = 0.

(4.2)

Taking inner product in (4.2) with π∗ F Z, we obtain gM (grad ln λ, Z)λ2 gM (F Z, F Z) + gM (grad ln λ, F Z)λ2 gM (Z, F Z)

(4.3)

−gM (Z, F Z)λ2 gM (grad ln λ, F Z) = 0. From (4.3), λ is a constant on (μ). On the other hand, for U, V ∈ (kerπ∗ ), from Lemma 2.2 we have (∇π∗ )(F U, F V ) = F U (ln λ)π∗ F V + F V (ln λ)π∗ F U − gM (F U, F V )π∗ (grad ln λ). Again, if π is a horizontally homothetic map, then (∇π∗ )(F U, F V ) = 0. Conversely, if (∇π∗ )(F U, F V ) = 0, putting U instead of V in the above equation, we derive 2F U (ln λ)π∗ F U − gM (F U, F U )π∗ (grad ln λ) = 0.

(4.4)

Since π is a conformal submersion, taking inner product in (4.4) with π∗ F U , we have gM (F U, F U )λ2 gM (grad ln λ, F U ) = 0.

M. A. Akyol

From the above equation, λ is a constant on (F kerπ∗ ). Thus, λ is a constant on ((kerπ∗ )⊥ ). (b) For U, V ∈ (D1 ), using (2.2), (2.9) and (2.5), we have (∇π∗ )(U, V ) = −π∗ (F (TU F V + ∇ˆ U F V )). Using (3.3) and (3.4) in the above equation, we obtain (∇π∗ )(U, V ) = −π∗ (B TU F V + C TU F V + φ ∇ˆ U F V + ω∇ˆ U F V ). Since B TU F V + φ ∇ˆ U F V ∈ (kerπ∗ ), we derive (∇π∗ )(U, V ) = −π∗ (C TU F V + ω∇ˆ U F V ). Then, since π is a linear isomorphism between (kerπ∗ )⊥ and T B, (∇π∗ )(U, V ) = 0 if and only if C TU F V + ω∇ˆ U F V = 0. (c) For U ∈ (kerπ∗ ) and W ∈ (D2 ), using (2.2) and (2.9), we have M

(∇π∗ )(U, W ) = ∇Uπ π∗ W − π∗ (∇U W ) M

= −π∗ (F ∇U F W ). Then from (2.6), we arrive at M

(∇π∗ )(U, W ) = −π∗ (F (TU F W + H∇U F W )). Using (3.3) and (3.4) in the above equation, we obtain M

M

(∇π∗ )(U, W ) = −π∗ (φTU F W + ωTU F W + BH∇U F W + CH∇U F W ). M

Since φTU F W + BH∇U F W ∈ (kerπ∗ ), we derive M

(∇π∗ )(U, W ) = −π∗ (ωTU F W + CH∇U F W ). Then, since π is a linear isomorphism between (kerπ∗ )⊥ and T B, (∇π∗ )(U, W ) = 0 if M and only if ωTU F W + CH∇U F W = 0. (d) For U ∈ (kerπ∗ ) and X ∈ ((kerπ∗ )⊥ ), using (2.2), (2.9) and (2.6), we have M

(∇π∗ )(U, X) = ∇Uπ π∗ X − π∗ (∇U X) M

= −π∗ (F ∇U F X). Then from (3.4), (2.5), and (2.6), we arrive at M (∇π∗ )(U, X) = −π∗ (F (TU B X + ∇ˆ U B X) + F (TU C X + H∇U C X)).

Using (3.3) and (3.4) in the above equation, we obtain (∇π∗ )(U, X) = −π∗ (B TU B X + C TU B X + φ ∇ˆ U B X + ω∇ˆ U B X M

M

+φTU C X + ωTU C X + BH∇U C X + CH∇U C X). M Since B (TU B X + H∇U C X) + φ(∇ˆ U B X + TU C X) ∈ (kerF∗ ), we derive

(∇π∗ )(U, X) = −π∗ (C (TU B X + H∇U C X) + ω(∇ˆ U B X + TU C X)). M

Conformal Semi-˙Invariant Submersions...

Then, since π is a linear isomorphism between (kerπ∗ )⊥ and T B, (∇π∗ )(U, X) = 0 if and M only if C (TU B X + H∇U C X) + ω(∇ˆ U B X + TU C X) = 0, which implies that TU B X + M H∇U C X ∈ (F D2 ) and ∇ˆ U B X + TU C X ∈ (D1 ). Hence, the proof is complete. Acknowledgments

The author is grateful to the referee for his/her valuable comments and suggestions.

References 1. Akyol, M.A.: Conformal anti-invariant submersions from almost product Riemannian manifolds onto Riemannian manifolds, Submitted (2016) 2. Akyol, M.A., S.ahin, B.: Conformal anti-invariant submersions from almost Hermitian manifolds. Turk. J. Math. 40, 43–70 (2016) 3. Akyol, M.A., S.ahin, B.: Conformal semi-invariant submersions. Communications in Contemporary Mathematics, Inpress. doi:10.1142/S02191997165001151650011 4. Baird, P., Wood, J.C.: Harmonic Morphisms Between Riemannian Manifolds. London Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press Oxford (2003) 5. Fuglede, B.: Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier (Grenoble) 28, 107–144 (1978) 6. Falcitelli, M., Ianus, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific, River Edge NJ (2004) 7. Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967) 8. Gundmundsson, S., Wood, J.C.: Harmonic morphisms between almost Hermitian manifolds. Boll. Un. Mat. Ital. B. 11(2), 185–197 (1997) 9. G¨und¨uzalp, Y.: Anti-invariant Riemannian submersions from almost product Riemannian manifolds. Mathematical Science and Applications E-notes 1(1), 58–66 (2013) 10. G¨und¨uzalp, Y.: Slant submersions from almost product Riemannian manifolds. Turk. J. Math. 37, 863– 873 (2013) 11. Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19, 215–229 (1979) 12. O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 458–469 (1966) 13. Park, K.S.: h-semi-invariant submersions. Taiwan. J. Math. 16(5), 1865–1878 (2012) 14. S¸ahin, B.: Anti-invariant Riemannian submersions from almost Hermitian manifolds. Central European J. Math 3, 437–447 (2010) 15. S¸ahin, B.: Semi-invariant Riemannian submersions from almost Hermitian manifolds. Canad. Math. Bull. 56, 173–183 (2013) 16. Watson, B.: Almost Hermitian submersions. J. Differ. Geom. 11(1), 147–165 (1976) 17. Tas¸tan, H.M.: Anti-holomorphic semi-invariant submersions from K¨ahlerian manifolds. arXiv:1404.2385v1 [math.DG] 18. Urakawa, H.: Calculus of variations and harmonic maps. Am. Math. Soc., 132 (1993) 19. Yano, K., Kon, M.: Structures on Manifolds. World Scientific, Singapore (1984)