Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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1 Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica Kanak Kanti Baishya; Partha Roy Chowdhury; Josef Mikeš; Patrik Peška On Almost Generalized Weakly Symmetric Kenmotsu Manifolds Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica Vol. 55 (206) No Persistent URL: Terms of use: Palacký University Olomouc Faculty of Science 206 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Acta Univ. Palacki. Olomuc. Fac. rer. nat. Mathematica 55 2 (206) 5 5 On Almost Generalized Weakly Symmetric Kenmotsu Manifolds Kanak Kanti BAISHYA Partha Roy CHOWDHURY 2 Josef MIKEŠ 3a PatrikPEŠKA 3b Department of Mathematics Kurseong College Dowhill Road Kurseong Darjeeling West Bengal India kanakkanti.kc@gmail.com 2 Department Of Mathematics Shaktigarh Bidyapith(H.S) Darjeeling West Bengal India partha.raychowdhury8@gmail.com 3 Department of Algebra and Geometry Faculty of Science Palacky University 7. listopadu Olomouc Czech Republic a josef.mikes@upol.cz b patrik peska@seznam.cz (Received February 6 206) Abstract This paper aims to introduce the notions of an almost generalized weakly symmetric Kenmotsu manifolds and an almost generalized weakly Ricci-symmetric Kenmotsu manifolds. The existence of an almost generalized weakly symmetric Kenmotsu manifold is ensured by a non-trivial example. Key words: Almost generalized weakly symmetric Kenmotsu manifolds almost generalized weakly Ricci-symmetric Kenmotsu manifolds. 200 Mathematics Subject Classification: 53C5 53C25 Introduction The notion of a weakly symmetric Riemannian manifold has been initiatied by Tamássy and Binh [22]. In the spirit of [5] a weakly symmetric Riemannian manifold (M n g) issaidtobeann-dimensional almost weakly pseudo symmetric manifold (n >2) if its curvature tensor R of type (0 4) is not identically 5

3 6 K. K. Baishya P. R. Chowdhury J. Mikeš P. Peška zero and admits the identity ( X R)(YUVW) =[A (X)+B (X)] R(YUVW) + C (Y ) R(X U V W)+C (U) R(YXVW) + D (V ) R(YUXW)+D (W ) R(YUVX) (.) where A B C D are non-zero -forms defined by A (X) = g(x σ ) B (X) =g(x ϱ )C (X) =g(x π ) and D (X) =g(x ) for all X and R(YUVW) =g(r(yu)vw) being the operator of the covariant differentiation with respect to the metric tensor g. An n-dimensional Riemannian manifold of this kind is denoted by A(WPS) n -manifold. Keeping the tune of Dubey [8] we shall call a Riemannian manifold of dimension n an almost generalized weakly symmetric (which is abbreviated hereafter as A(GW S) n -manifold) if it admits the equation ( X R)(YUVW) =[A (X)+B (X)]R(YUVW)+C (Y )R(X U V W) + C (U)R(YXVW)+D (V )R(YUXW) + D (W )R(YUVX)+[A 2 (X)+B 2 (X)]Ḡ(YUVW) + C 2 (Y )Ḡ(X U V W)+C2(U)Ḡ(YXVW) + D 2 (V ) Ḡ(YUXW)+D 2(W )Ḡ(YUVX) (.2) where G(YUVW) =g(u V )g(yw) g(yv )g(u W ) and A i B i C i D i i = 2 are non-zero -forms defined by A i (X) =g(x σ i ) B i (X) =g(x ϱ i ) C i (X) =g(x π i ) D i (X) =g(x i ). The beauty of such A(GW S) n -manifold is that it has the flavour of (i) locally symmetric space in the sense of Cartan for A i = B i = C i = D i =0 (ii) recurrent space by Walker [24] for A 0 B i = C i = D i =0 (iii) generalized recurrent space by Dubey[8] for A i 0and B i = C i = D i =0 (iv) pseudo symmetric space by Chaki [4] for A = B = C = D 0and A 2 = B 2 = C 2 = D 2 =0 (v) semi-pseudo symmetric space in the sense of Tarafder et al. [23] for A = B C = D and A 2 = B 2 = C 2 = D 2 =0 (vi) generalized semi-pseudo symmetric space in the sense of Baishya [] for A = B C = D and A 2 = B 2 C 2 = D 2 =0 (vii) generalized pseudo symmetric space bybaishya[2] for A i = B i = C i = D i 0 (viii) almost pseudo symmetric space in the sprite of Chaki et al. [5] for B 0 A = C = D 0and A 2 = B 2 = C 2 = D 2 =0 (ix) almost generalized pseudo symmetric space in the sence of Baishya for B i 0 A i = C i = D i 0

4 On Almost Generalized Weakly Symmetric Kenmotsu Manifolds 7 (x) weakly symmetric space by Tamássy and Binh [22] for A 2 = B 2 = C 2 = D 2 =0. Our work is structured as follows. Section 2 is concerned with Kenmotsu manifolds and some known results. In section 3 we have investigated an almost generalized weakly symmetric Kenmotsu manifold and obtained some interesting results. Section 4 is concerned with an almost generalized weakly Riccisymmetric Kenmotsu manifold. Finally we have constructed an example of an almost generalized weakly symmetric Kenmotsu manifold. 2 Kenmotsu manifolds and some known results Let M be a n-dimensional connected differentiable manifold of class C -covered by a system of coordinate neighborhoods (U x h ) in which there are given a tensor field ϕ of type ( ) a cotravariant vector field ξ and a form η such that ϕ 2 X = X + η(x) ξ (2.) η(ξ) = ϕ ξ =0 η(ϕx) =0 (2.2) for any vector field X on M. Then the structure (ϕ ξ η) is called contact structure and the manifold M n equipped with such structure is said to be an almost contact manifold if there is given a Riemannian compatible metric g such that g(ϕx Y )= g(x ϕy ) g(x ξ) =η(x) (2.3) g(ϕx ϕy )=g(x Y ) η(x)η(y ) (2.4) for all vector fields X and Y thenwesaym is an almost contact metric manifold. An almost contact metric manifold M is called a Kenmotsu manifold if it satisfies [] ( X ϕ)y = g(x ϕy )ξ η(y )ϕ(x) (2.5) for all vector fields X and Y where is a Levi-Civita connection of the Riemannian metric. From the above it follows that X ξ = X η(x)ξ (2.6) ( X η)y = g(x Y ) η(x) η(y ) (2.7) In a Kenmotsu manifold the following relations hold ([7] [0]) R(X Y )ξ = η(x)y η(y )X (2.8) S(X ξ) = (n ) η(x) (2.9) R(X ξ)y = g(x Y )ξ η(y )X (2.0) R(ξX)Y = η(y )X g(x Y )ξ (2.) for any vector fields X Y Z where R is the Riemannian curvature tensor of the manifold.

5 8 K. K. Baishya P. R. Chowdhury J. Mikeš P. Peška 3 Almost generalized weakly symmetric Kenmotsu manifold A Kenmotsu manifold (M n g) is said to be an almost generalized weakly symmetric if it admits the relation (.) (n >2). Now contracting Y over W in both sides of (.) we get ( X S)(U V )=[A (X)+B (X)]S(U V )+C (U)S(X V ) + C (R(X U)V )+D (R(X V )U)+D (V )S(U X) +(n )[{A 2 (X)+B 2 (X)}g(U V )+C 2 (U)g(X V ) + D 2 (V )g(u X)] + C 2 (G(X U)V )+D 2 (G(X V )U). (3.) In consequence of (2.8) (2.9) and (2.0) for V = ξ the above equation yields ( X S)(U ξ) = (n )[A (X)+B (X)]η(U) (n 2)C (U)η(X) + D (ξ)s(u X) η(u) C (X) η(u)d (X) + g(x U)D (ξ)+(n )[{A 2 (X)+B 2 (X)η(U) + C 2 (U)η(X)+D 2 (ξ)g(u X)] + η(u)c 2 (X) η(x)c 2 (U)+η(U)D 2 (X) g(u X)D 2 (ξ). (3.2) Again replacing V by ξ in the following identity ( X S)(U V )= X S(U V ) S( X U V ) S(U X V ) (3.3) and then making use of (2.) (2.6) (2.9) we find Now using (3.4) in (3.2) we have ( X S)(U ξ) = (n )g(x U) S(U X). (3.4) (n )g(x U) S(U X) = (n )[{A (X)+B (X)}η(U)] (n 2)C (U)η(X)+D (ξ)s(u X) η(u)c (X)+g(X U)D (ξ) η(u)d (X) +(n )[{A 2 (X)+B 2 (X)}η(U)+C 2 (U)η(X)+D 2 (ξ)g(u X)] + η(u)c 2 (X) η(x)c 2 (U)+η(U)D 2 (X) g(u X)D 2 (ξ) (3.5) which leaves [A (ξ)+b (ξ)+c (ξ)+d (ξ)] = [A 2 (ξ)+b 2 (ξ)+c 2 (ξ)+d 2 (ξ)] (3.6) for X = U = ξ. In particular if A 2 (ξ) =B 2 (ξ) =C 2 (ξ) =D 2 (ξ) =0 formula (3.6) turns into A (ξ)+b (ξ)+c (ξ)+d (ξ) =0. (3.7) This leads to the following

6 On Almost Generalized Weakly Symmetric Kenmotsu Manifolds 9 Theorem. In an almost generalized weakly symmetric Kenmotsu manifold (M n g) n>2 the relation (3.6) hold good. In a similar manner we can have (n )g(x V ) S(VX) = (n )[A (X)+B (X)]η(V ) (n 2)D (V )η(x)] + C (ξ)s(x V )+g(x V )C (ξ) η(v )C (X) η(v )D (X) +(n )[{A 2 (X)+B 2 (X)}η(V )+C 2 (ξ)g(x V ) + D 2 (V )η(x)] + η(v )C 2 (X) g(x V )C 2 (ξ) + η(v )D 2 (X) η(x)d 2 (V ) (3.8) Now putting V = ξ in (3.8) and using (2.) (2.9) we obtain (n )[A (X)+B (X)] + C (X)+D (X) +(n 2)[C (ξ)+d (ξ)]η(x) =[(n ){A 2 (X)+B 2 (X)} +(n 2){C 2 (ξ)+d 2 (ξ)}η(x)] + C 2 (X)+D 2 (X) (3.9) Putting X = ξ in (3.8) and using (2.) (2.2) (2.9) we obtain (n )[A (ξ)+b (ξ)+c (ξ)]η(v )+(n 2)D (V )+η(v)d (ξ) =(n )[{A 2 (ξ)+b 2 (ξ)+c 2 (ξ)}η(v )+D 2 (V )] + η(v )D 2 (ξ) D 2 (V ) (3.0) Replacing V by X in the above equation and using (3.6) we get D (X) D (ξ)η(x) =D 2 (X) D 2 (ξ)η(x) (3.) In view of (3.6) (3.9) and (3.) we get C (X) C (ξ)η(x) =C 2 (X) C 2 (ξ)η(x). (3.2) Subtracting (3.) (3.2) from (3.9) we get [A (X)+B (X)] + [C (ξ)+d (ξ)]η(x) = {A 2 (X)+B 2 (X)} + {C 2 (ξ)+d 2 (ξ)}η(x)] (3.3) Again adding (3.) (3.2) and (3.3) we get A (X)+B (X)+C (X)+D (X) =[A 2 (X)+B 2 (X)+C 2 (X)+D 2 (X)]. (3.4) Next in view of A 2 = B 2 = C 2 = D 2 =0 the relation (3.4) yields A (X)+B (X)+C (X)+D (X) =0. (3.5) This motivates us to state the followings

7 0 K. K. Baishya P. R. Chowdhury J. Mikeš P. Peška Theorem 2. In an almost generalized weakly symmetric Kenmotsu manifold (M n g) n>3 the sum of the associated -forms is given by (3.4). Theorem 3. There does not exist a Kenmotsu manifold which is (i) recurrent (ii) generalized recurrent provided the -forms are collinear (iii) pseudo symmetric (iv) generalized semi-pseudo symmetric provided the -forms are collinear (v) generalized almost-pseudo symmetric provided the -forms are collinear. 4 Almost generalized weakly Ricci-symmetric Kenmotsu manifold A Kenmotsumanifold (M n g)(n >3) is said to be almost generalized weakly Ricci-symmetric if there exist -forms Āi Bi C i and D i which satisfy the condition ( X S)(U V ) =[Ā(X)+ B (X)]S(U V )+ C (U)S(X V )+ D (V )S(U X) +[Ā2(X)+ B 2 (X)]g(U V )+ C 2 (U)g(X V )+ D 2 (V )g(u X). (4.) Putting V = ξ in (4.) we obtain ( X S)(U ξ) =[Ā(X)+ B (X)](n 2)η(U)+ C (U)(n 2)η(X)+ D (ξ)s(u X) +[Ā2(X)+ B 2 (X)]η(U)+ C 2 (U)η(X)+ D 2 (ξ)g(u X). (4.2) In view of (3.4) the relation (4.2) becomes (n )g(x U) S(U X) = (n )[{Ā(X)+ B (X)}η(U)+ C (U)η(X)] + D (ξ)s(u X) +[Ā2(X)+ B 2 (X)]η(U)+ C 2 (U)η(X)+ D 2 (ξ)g(u X). (4.3) Setting X = U = ξ in (4.3) and using (2.) (2.2) and (2.9) we get (n )[Ā(ξ)+ B (ξ)+ C (ξ)+ D (ξ)] =[Ā2(ξ)+ B 2 (ξ)+ C 2 (ξ)+ D 2 (ξ)]. (4.4) Again putting X = ξ in (4.3) we get (n )[{Ā(ξ)+ B (ξ)+ D (ξ)}η(u)+ C (U)] =[Ā2(ξ)+ B 2 (ξ)+ D 2 (ξ)]η(u)+ C 2 (U). (4.5) Setting U = ξ in (4.3) and then using (2.) (2.2) and (2.9) we obtain (n )[{Ā(X)+ B (X)} + { C (ξ)+ D (ξ)}η(x)] =[Ā2(X)+ B 2 (X)] + C 2 (ξ)η(x)+ D 2 (ξ)η(x). (4.6)

8 On Almost Generalized Weakly Symmetric Kenmotsu Manifolds Replacing U by X in (4.5) and adding with (4.6) we have (n )[Ā(X)+ B (X)+ C (X)] [Ā2(X)+ B 2 (X)+ C 2 (X)] = (n )[Ā(ξ)+ B (ξ)+ C (ξ)+ D (ξ)]η(x) +[Ā2(ξ)+ B 2 (ξ)+ C 2 (ξ)+ D 2 (ξ)]η(x) (n ) D (ξ)η(x)+ D 2 (ξ)η(x). (4.7) In consequence of (4.4) the above equation becomes (n )[Ā(X)+ B (X)+ C (X)] + (n ) D (ξ)η(x). =[Ā2(X)+ B 2 (X)+ C 2 (X)] + D 2 (ξ)η(x) (4.8) Next putting X = U = ξ in (4.) we get (n )[Ā(ξ)+ B (ξ)+ C (ξ)]η(v )+(n ) D (V ) =[Ā2(ξ)+ B 2 (ξ)+ C 2 (ξ)]η(v )+ D 2 (V ) (4.9) Replacing V by X in (4.9) and adding with (4.8) we obtain (n )[Ā(X)+ B (X)+ C (X)+ D (X)] (n )[Ā(ξ)+ B (ξ)+ C (ξ)+ D (ξ)]η(v ) =[Ā2(X)+ B 2 (X)+ C 2 (X)+ D (X)] +[Ā2(ξ)+ B 2 (ξ)+ C 2 (ξ)+ D 2 (ξ)]η(v ). (4.0) By virtue of (4.4) the above equation becomes (n )[Ā(X)+ B (X)+ C (X)+ D (X)] =[Ā2(X)+ B 2 (X)+ C 2 (X)+ D (X)]. (4.) This leads to the followings Theorem 4. In an almost generalized weakly Ricci symmetric Kenmotsu manifold Theorem 5. (M n g) n>2 the sum of the associated -forms are related by (4.). 5 Example of an A(GWS) 3 Kenmotsu manifold (see [7] page 2 22) Let M 3 (ϕ ξ η g) be a Kenmotsu manifold (M 3 g) with a ϕ-basis e = e z x e 2 = e z y e 3 = z.

9 2 K. K. Baishya P. R. Chowdhury J. Mikeš P. Peška Then from Koszul s formula for Riemannian metric g we can obtain the Levi- Civita connection as follows e e 3 = e e e 2 =0 e e = e 3 e2 e 3 = e 2 e2 e 2 = e 3 e2 e =0 e3 e 3 =0 e3 e 2 =0 e3 e =0. Using the above relations one can easily calculate the non-vanishing components of the curvature tensor R (up to symmetry and skew-symmetry) R(e e 3 e e 3 )=R(e 2 e 3 e 2 e 3 )==R(e e 2 e e 2 ). Since {e e 2 e 3 } forms a basis any vector field X Y U V χ(m) can be written as X = a i e i Y = b i e i U = c i e i V = d i e i R(X Y U V )=(a b 2 a 2 b )(c d 2 c 2 d )+(a b 3 a 3 b )(c d 3 c 3 d )+(a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d 2 )=T (say) R(e YUV)=b 3 (c d 3 c 3 d )+b 2 (c d 2 c 2 d )=λ (say) R(e 2 YUV)=b 3 (c 2 d 3 c 3 d 2 ) b (c d 2 c 2 d )=λ 2 (say) R(e 3 YUV)=b (c 3 d c d 3 )+b 2 (c 3 d 2 c 2 d 3 )=λ 3 (say) R(X e UV)=a 3 (c d 3 c 3 d )+a 2 (c d 2 c 2 d )=λ 4 (say) R(X e 2 UV)=a 3 (c 2 d 3 c 3 d 2 )+a (c 2 d c d 2 )=λ 5 (say) R(X e 3 UV)=a (c 3 d c d 3 )+a 2 (c 3 d 2 c 2 d 3 )=λ 6 (say) R(X Y e V)=d 3 (a b 3 a 3 b )+d 2 (a b 2 a 2 b )(= λ 7 (say) R(X Y e 2 V)=d 3 (a 2 b 3 a 3 b 2 )+d (a 2 b a b 2 )=λ 8 (say) R(X Y e 3 V)=d (a 3 b a b 3 )+d 2 (a 3 b 2 a 2 b 3 )=λ 9 (say) R(X Y U e )=c 3 (a b 3 a 3 b )+c 2 (a b 2 a 2 b )=λ 0 (say) R(X Y U e 2 )=c 3 (a 2 b 3 a 3 b 2 )+c (a 2 b a b 2 )=λ (say) R(X Y U e 3 )=c (a 3 b a b 3 )+c 2 (a 3 b 2 a 2 b 3 )=λ 2 (say) G(X Y U V )=(b c + b 2 c 2 b 3 c 3 )(a d + a 2 d 2 a 3 d 3 ) (a c + a 2 c 2 a 3 c 3 )(b d + b 2 d 2 b 3 d 3 )=T 2 (say) G(e YUV)=(b 2 c 2 b 3 c 3 )d (b 2 d 2 b 3 d 3 )c = ω (say) G(e 2 YUV)=(b c b 3 c 3 )d 2 (b d b 3 d 3 )c 2 = ω 2 (say) G(e 3 YUV)=(b c b 2 c 2 )d 3 (b d b 2 d 2 )c 3 = ω 3 (say) G(X e UV)=(a 2 d 2 a 3 d 3 )c (a 2 c 2 a 3 c 3 )d = ω 4 (say) G(X e 2 UV)=(a d a 3 d 3 )c 2 (a c a 3 c 3 )d 2 = ω 5 (say) G(X e 3 UV)=(a d a 2 d 2 )c 3 (a c a 2 c 2 )d 3 = ω 6 (say)

10 On Almost Generalized Weakly Symmetric Kenmotsu Manifolds 3 G(X Y e V)=(a 2 d 2 a 3 d 3 )b (b 2 d 2 b 3 d 3 )a = ω 7 (say) G(X Y e 2 V)=(a d a 3 d 3 )b 2 (b d b 3 d 3 )a 2 = ω 8 (say) G(X Y e 3 V)=(b d b 2 d 2 )a 3 (a d a 2 d 2 )b 3 = ω 9 (say) G(X Y U e )=(b 2 c 2 b 3 c 3 )a (a 2 c 2 a 3 c 3 )b = ω 0 (say) G(X Y U e 2 )=(b c b 3 c 3 )a 2 (a c a 3 c 3 )b 2 = ω (say) G(X Y U e 3 )=(b c b 2 c 2 )a 3 (a c + a 2 c 2 )b 3 = ω 2 (say) and the components which can be obtained from these by the symmetry properties. Now we calculate the covariant derivatives of the non-vanishing components of the curvature tensor as follows ( e R)(X Y U V )= = a λ 3 + a 3 λ 2 b λ 6 + b 3 λ 5 c λ 9 + c 3 λ 8 d λ 2 + d 3 λ ( e2 R)(X Y U V )= = a 2 λ 3 + a 3 λ 2 b 2 λ 6 + b 3 λ 5 c 2 λ 9 + c 3 λ 8 d 3 λ d 2 λ 2 ( e3 R)(X Y U V )=0 For the following choice of the the one forms A (e )= a 3λ 2 a λ 3 T B (e )= b 3λ 5 b λ 6 T A 2 (e )= c 3λ 8 c λ 9 T 2 B 2 (e )= d 3λ d λ 2 T 2 A (e 2 )= a 3λ 2 a 2 λ 3 T B (e 2 )= b 3λ 5 b 2 λ 6 T A 2 (e 2 )= c 3λ 8 c 2 λ 9 T 2 B 2 (e 2 )= d 3λ d 2 λ 2 T 2 A (e 3 )= e2z (a b 2 a 2 b )c d 2 T B (e 3 )= e2z (a b 2 a 2 b )c d 2 T C (e 3 )= C 2 (e 3 )= a 3 λ 3 + b 3 λ 6 a 3 θ 3 + b 3 θ 6 D (e 3 )= D 2 (e 3 )= c 3 λ 9 + d 3 λ 2 c 3 θ 9 + d 3 θ 2 A 2 (e 3 )= α2 e 2z (a b 2 a 2 b )c d 2 T 2 B 2 (e 3 )= α2 e 2z (a b 2 a 2 b )c 2 d T 2

11 4 K. K. Baishya P. R. Chowdhury J. Mikeš P. Peška one can easily verify the relations ( ei R)(X Y U V )=[A (e i )+B (e i )]R(X Y U V ) + C (X)R(e i YUV)+C (Y )R(X e i UV) + D (U)R(X Y e i V)+D (V )R(X Y U e i ) +[A 2 (e i )+B 2 (e i )]G(X Y U V ) + C 2 (X)G(e i YUV)+C 2 (Y )G(X e i UV) + D 2 (U)G(X Y e i V)+D 2 (V )G(X Y U e i ) for 2 3. From the above we can state that Theorem 6. There exixt a Kenmotsu manifold (M 3 g) which is an almost generalized weakly symmetry Kenmotsu manifold. References [] Baishya K. K.: On generalized semi-pseudo symmetric manifold. submitted. [2] Baishya K. K.: On generalized pseudo symmetric manifold. submitted. [3] Baishya K. K.: On almost generalized pseudo symmetric manifolds. submitted. [4] Chaki M. C.: On pseudo Ricci symmetric manifolds. Bulg. J. Physics5 (988) [5] Chaki M. C. Kawaguchi T.: On almost pseudo Ricci symmetric manifolds. Tensor68 (2007) 0 4. [6] Deszcz R. Glogowska M. Hotlos M. Senturk Z.: On certain quasi-einstein semisymmetric hypersurfaces. Annales Univ. Sci. Budapest. Eotovos Sect. Math. 4 (998) [7] Dey S. K. Baishya K. K.: On the existence of some types on Kenmotsu manifolds. Universal Journal of Mathematics and Mathematical Sciences 3 2 (204) [8] Dubey R. S. D.: Generalized recurrent spaces. Indian J. Pure Appl. Math. 0 (979) [9] Janssens D. Vanhecke L.: Almost contact structures and curvature tensors. Kodai Math. J. 4 (98) 27. [0] June J. B. Dey U. C. Pathak G.: On Kenmotsu manifolds. J.KoreanMath.Soc.42 3 (2005) [] Kenmotsu K.: A class of almost contact Riemannian manifolds. Tohoku Mathematical Journal 24 (972) [2] Hinterleitner I. Mikeš J.: Geodesic mappings onto Weyl manifolds. J. Appl. Math. 2 (2009) [3] Mikeš J.: Geodesic mappings of semisymmetric Riemannian spaces. Archives at VINITI Odessk. Univ. Moscow 976. [4] Mikeš J.: On geodesic mappings of 2-Ricci symmetric Riemannian spaces. Math.Notes 28 (98) [5] Mikeš J.: On geodesic mappings of Einstein spaces. Math.Notes28 (98) [6] Mikeš J.: On geodesic mappings of m-symmetric and generally semi-symmetric spaces. Russ. Math (992) [7] Mikeš J.: On geodesic and holomorphic-projective mappings of generalized m-recurrent Riemannian spaces. Sib.Mat.Zh.33 5 (992)

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