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1 I nternational 9, S etember Some Classes of almost r- C ontact Riemannian a nd Kenmotsu Manifold Manifolds G eeta Verma, V irendra Nath Pathak A bstract Certain classes of almost r- Riemannian Manifolds, viz., almost Kenmotsu, nearly Kenmotsu, Quasi-Kenmotsu and secial r-c ontact metric Manifolds are defined and obtained some roerties of these s. Also, it has been shown that the structure vector field of the almost r- metric structure,,, G) ( i s not a Killing vector field on a nearly K enmotsu vector field I ndex Terms almost r- Riemannian, Sasakian, Kenmotsu Manifold, Killing vector f ield. I. IN TROUCTION The study of odd dimensional s with r- and almost r-c ontact structures was initiated by Boothby and Wang in 1958 rather from toological oint of view. Sasaki and Hatakeyama reinvestigated them using tensor calculus in1961.almost r- metric structures and Sasakian structures viz., almost Sasakian, nearly Sasakian etc., were roosed by Sasaki [5] in 1960 and 1965 resectively. Later, Kenmotsu [3] defined a class of almost r- Riemannian, called Kenmotsu, similar in arallel to Sasakian in In this aer, we defined almost Kenmotsu, nearly Kenmotsu, Quasi- Kenmotsu and secial r- metric Manifolds. The relation among these s has been obtained and studied some roerties of t hese s. Manuscrit received Se 11, Geet a Verma, eartment of Mathematics, Shri Ramswaroo Memorial G rou of Professional Colleges, Tewariganj, Faizabad Road, Lucknow V irendra Nath Pathak, Shri Ramswaroo Memorial University. Lucknow-eva Road , Uttar Pradesh 29

2 S ome Classes of almost r-c ontact Riemannian Manifolds and Kenmotsu Manifold 1. Peliminaries M M 2mr (,,, G Let b e a ( 2 r) m imensional almost d r- metric with structure tensors ( ) where is a tensor of tye (1,1), is a vector field, is a 1- form and G is the associated Riemannian metric on M. ( 1.1) 2 T hen definition, we have ( ),, ), ) ( ) ( ) 0 The fundamental 2- form i s defined by: ( 1.2) (, ), ) w here we ut. If M is an almost r- c ontact metric, we have ( 1.3) ( )(, ) ( )( ) ( 1.4) )(, ) (, ) ( )( ) ( ) )( ( ) W here is the Riemannian connection determined by the metric G. On the almost r-c ontact metric if further we have )( ) ( )( ), ) ( 1.5) I t is called Kenmotsu Manifold [3]. F rom (1. 1 ) and (1.5), we get ( ) ( 1.6) T hen from equations (1.1) and (1.6), we get ( 1.7) )( ), ) ( ) ( ), ) S imilarly from (1.5) we also have ( 1.8) )( )( ) )(, ) 2[ ( ) ( ) ) (, ) An almost r- c ontact structure is said to be Normal if N (, ) v anishes, where N def (, ) N d (, ) (, ) ( 1.9) H ere, (, ) is known as the Nijenhuis tensor of. N 2. Almost Kenmotsu and S-r- c ontact metric s: efinition (2.1): An almost r-c ontact metric d 0. hen the M is called an Almost Kenmtsu (or) r M on which there exists a function f such that T - c ontact metric. From the above definition, if is an affine connection on r-c ontact metric, we have ( 2.1) (a) )( ) ( )( ) [ T (, )] 0 W here T is the torsion tensor of. I f is symmetric, then from (2.1)(a), we have d f if 30

3 I nternational 9, S etember ( 2.1) (b) )( ) )( ) 0 N ote that, in the sequel, we shall always take as a Rie mannian connection in this aer. efinition (2.2): An almost Kenmotsu ( r- c ontact metric ) on which if the ( 2.2) ( )( ) ( )( ) 2, ) condition Is satisfied, then it is called a secial r-c ontact metric or in short S-r- c ontact metric. Therefore, from (2.1)(b) and (2.2), for an S-r- c ontact metric, we get ( 2.3) ( )( ) ( )( ), ) (, ) Theorem (2.1): In an almost r- metric, if m etric. On the almost r- c ontact metric, we have ( ) ) ( G G ( ), ), ) S imilarly, we see that )( ) ) is satisfied, it is an S-r- O n adding and subtracting the above two values, we get both (2.2) and (2.3) resectivel y, which roves the theorem. Preosition (2.1): In an S-r- c ontact metric, we get ( ) ( 2.4) P roof: T he equation (2.3) is equivalent to (2.4) Corollary (2.1): The following holds on S-r- c ontact metric : ( 2.5) )( ), ) P roof: T aking the covariant differentiation of (, ) 0, w e ge Theorem (2.2) : ( 2.6) t )( ) ( ) ( ), ) In an S-r- c ontact metric, we have ( i) )(, ) ', ) ( ii) )( ), ) he e quatio ( )( i mlies )( )( )(, )(, [, ], T n U sing (1.5), the above equation imlies ( ), ( ) ', ) W hich can also be written a (2.6) (i) and hence, also we have (2.6) (ii). Theorem (2.3) : On an S-r- c ontact metric, the condition ( 2.7) )(, ) ( ) ( ), I s equivalent to the condition ( 2.8) )(, ) ', ) 0 F rom (2.6) and Barring and (2.7) we get )(, ) ', ) 0 ( ) ( ), i n the above equation, we get (2.8). Again, barring and in (2.8), we get 31

4 S ome Classes of almost r-c ontact Riemannian Manifolds and Kenmotsu Manifold )(, ) ( )( )(, ) ( ) )( ) U sing (2.5), the above equation gives (2.7). 3. Quasi Kenmotsu : efinition (3.1): An almost r-c ontact metric is said to b e Quasi Kenmotsu if ( 3.1) )( )( ) )(, ) 0 T heorem (3.1): The necessary and sufficient condition for a Quasi-K enmotsu to be Normal is ( 3.2) ( )(, ) ( )( )( ). P roof: W e know that the necessary and sufficien t condition for a Quasi Kenmotsu which is an almost r- to be normal is N 0. That is, N (, ) d (, ) 0, W hich is reresented as )(, ) ( )(, ) ( )(, ) ( )(, ) (, ) d ( 0 u sing the equations (1.8),(1.4) and ( 2.3) the above equation gives (3.2). 4. Kenmotsu : efinition (4.1): An S-r- c ontact metric on which the equation (2.8) holds is called a Kenmotsu. T heorem (4.1): A normal S-r- c ontact metric is Kenmotsu. P roof: B y N (, ) 0, w e have )(, ) ( )(, ) ( )(, ) ( )(, ) (, ) d ( 0 U sing the equations (2.1), (1.8),(1.4) W hich also and (2.3) resectively, the above equation gives )(, ) ( ) ( ) P G (, imlies )(, ) G ( ) ). T his shows that the is of Kenmotsu tye. Hence, is a Normal S- r c ontact metric. 5. Nearly K enmotsu : efinition (5.1) : An almost r-c ontact metric on which ( 5.1) )( ) ( )( ) ( )( ) ( )( ) I s satisfied, is called a nearly Kenmotsu. T heorem (5.1): On nearly Kenmotsu i s not a Killing vector field. P roof: O erate (5.1) with G and ut, w e get )(, )(, ( ) U sing (1.3) and barring in this equation, we see that ( 5.2) )( )(, ( ) Consequently, we have an alternate definition of Kenmotsu is given as: A Kenmotsu [ ( )( ) ( )( )] [( )(, )(, ] 2(, U se of (1.4) in this equation yields (5.3) [( )( ) ( )( )] )( ) ( Writing for in (5.2), we get )( 0 w hich roves the theorem. )( ( ) 2(, By 32

5 I nternational T heorem (5.2): A normal nearly Kenmotsu is Kenmotsu. P roof: Since we have N 0, it holds that [, ] [, ] [, ] [, ] d (, ) 0 Oerating the above by i mlies ( 5.4) [ ) ) ] d (, ) 0 B arring in (5.1), and on oeration with, w e get ( 5.5) ) ( ). S imilarly, we see that ( 5.6) ) ( ) Using (5.5) and (5.6), equation (5.4) assumes the 9, S etember form )(, ) ( )(, ) d (, ) 0. equation yields d (, ) which shows that the is of almost Kenmotsu. By equation Use of (1.3) in this 0 ( 2.2) it is of S-r- c ontact metric and therefore the result follows from theorem (4.1). T heorem (5.3): A which is of nearly Kenmotsu and Quasi Kenmotsu is of Kenmotsu. P roof: O n a Nearly Kenmotsu, we have A dding the ( 5.7) F rom (1.5) we have )(, ) ( )(, ) G ( ), above equation to (1.8), we get 2 )( )(, ) 3 ( ), ( ) ) 2 ( ) ( 5.8) )( ) ( )(, ) ( ) ) (, ) H ence from (5.7) w hich roves and (5.8) we have )(, ) G ( ) )( ) ( )( ), ) I t shows that the is Kenmotsu. R EFERENCES [ 1].E. Blair, Contact s in Riemannian geometry, L ecture notes in Math. 509, Sringer Verlag, New ork. [ 2] Giusee Occhino, Vertical exterior concurrent vector field on Kenmotsu, Seminarberichte (Berlin), 39 ( 1990) [ 3] K.Kenmotsu, A class of Almost Riemannian s, Tohoku Math. Journal, 24 (1972) [ 4] R.S. Mishra, Structures on differentiable and their alications, ( 1984). Chandra Prakashan, Allahabad, India [ 5] S. Sasaki, On differentiable s with certain s tructures which are closely related to almost structures, Tohoku Math Journal, 12 (1960) [ 6] B.B. Sinha, K. L. Prasad, A class of Almost ara metric, Bulletin of the Calcutta Mathematical Society, 87 (1995)