The Schouten-van Kampen affine connections adapted to almost (para)contact metric structures (the work in progress) Zbigniew Olszak

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1 The Schouten-van Kampen affine connections adapted to almost (para)contact metric structures (the work in progress) Zbigniew Olszak Wroc law University of Technology, Wroc law, Poland XVII Geometrical Seminar September, 2012, Zlatibor, Serbia Zbigniew Olszak Schouten-van Kampen affine connections - 1/21

2 Affine connections adapted to a pair of distributions (1) M - a differentiable manifold, n = dim M 2. H, V - two complementary (smooth) distributions on M, TM = H V, H V = {0}, dim H = p, dim V = q, p + q = n, 1 p, q n 1. Df1. An affine connection is said to be adapted to (H, V) if ( ) X Y H for X TM, Y H, ( ) X Y V for X TM, Y V. In the above: ( ) (resp. ( )) means the parallelity of H (resp. V) with respect to. X h, X v - the projections of X TM onto H, V, resp. P - the almost product structure defined by PX = P(X h + X v ) := X h + X v for any X TM. A classical result states: An affine connection is adapted to (H, V) if and only if P = 0. Zbigniew Olszak Schouten-van Kampen affine connections - 2/21

3 Affine connections adapted to a pair of distributions (2) An affine connection is adapted to (H, V) if and only if ( X Y ) h = X Y h for any X, Y TM, or equivalently, ( X Y ) v = X Y v for any X, Y TM. In the case when dim H = n 1 and dim V = 1, it is possible to choose (at least locally) a vector field ξ and a linear form η such that H = ker η, V = span ξ, η(ξ) = 1. In this case, an affine connection is adapted to the pair (H, V) if and only if η(y ) X ξ = ( X η)(y )ξ for any X, Y TM, or equivalently, ( X η)(y ) = η(y )η( X ξ) and X ξ = η( X ξ)ξ for any X, Y TM. Zbigniew Olszak Schouten-van Kampen affine connections - 3/21

4 The Schouten-van Kampen affine connections Let be an affine connection on M. Df2. The Schouten-van Kampen affine connection associated to and adapted to (H, V) is defined by X Y := ( X Y h ) h + ( X Y v ) v, X, Y TM. B := - the second fundamental form, T, T - the torsion tensor fields of,, resp. We have, among others, (1) B = 0 = P = 0; (2) B is symmetric if and only if T = T. Zbigniew Olszak Schouten-van Kampen affine connections - 4/21

5 The Riemannian case, and dim H = n 1 (1) (M, g) - a pseudo-riemannian manifold (incl. Riemannian), n = dim M 2. H, V - two complementary and orthogonal distributions on M, V is non-null, and dim H = n 1, dim V = 1. Choose (at least locally) a vector field ξ and a linear form η such that H = ker η, V = span ξ, g(ξ, ξ) = ε 0 = ±1, η(ξ) = 1. Then, we have for any X TM, X ξ H, η(x ) = ε 0 g(x, ξ), X h = X η(x )ξ, X v = η(x )ξ, being the Levi-Civita connection. Zbigniew Olszak Schouten-van Kampen affine connections - 5/21

6 The Riemannian case, and dim H = n 1 (2) Th1. The Schouten-van Kampen affine connection associated to the Levi-Civita connection and adapted to the pair (H, V) is given by X Y = X Y η(y ) X ξ + ( X η)(y )ξ. Therefore, B(X, Y ) = η(y ) X ξ ( X η)(y )ξ, T (X, Y ) = η(x ) Y ξ η(y ) X ξ + 2dη(X, Y )ξ. For the horizontal and vertical parts of B, B = B h + B v, B h = ( ξ) η, B v = ( η) ξ. For the horizontal and vertical parts of T, T = T h + T v, T h = 2η ( ξ), T v = 2dη ξ. Zbigniew Olszak Schouten-van Kampen affine connections - 6/21

7 The Riemannian case, and dim H = n 1 (3) In the rest of this section, denotes the Schouten-van Kampen connection associated to the Levi-Civita connection and adapted to the pair (H, V). Th2. (a) ξ is a Killing vector field if and only if B v is a skew-symmetric vector valued 2-form; (b) η is a closed linear form if and only if T v = 0, or equivalently, B v is a symmetric vector valued 2-form. Th3. For the Schouten-van Kampen affine connection, ξ = 0, g = 0, η = 0. Th4. The Schouten-van Kampen affine connection is just the only one affine connection, which is metric and its torsion is of the form T = 2η ( ξ) + 2dη ξ. Zbigniew Olszak Schouten-van Kampen affine connections - 7/21

8 The Riemannian case, and dim H = n 1 (4) Th5. The following conditions are equivalent (a) the distribution H is involutive; (b) T (X, Y ) = 0 for any X, Y H; (c) T = η σ for a certain vector valued 1-form σ; (d) η T = 0. Th6. The curvature operators R and R of the connections and are related by R(X, Y )Z = (R(X, Y )Z h ) h Hint. + ( Y η)(z) X ξ ( X η)(z) Y ξ. (R(X, Y )Z h ) h = R(X, Y )Z η(r(x, Y )Z)ξ η(z)r(x, Y )ξ. Zbigniew Olszak Schouten-van Kampen affine connections - 8/21

9 The Riemannian case, and dim H = n 1 (5) Cor1. R(X, Y )Z H for any X, Y, Z TM. Define the Ricci curvature tensors, Rc(Y, Z) = Tr{X R(X, Y )Z}, Rc(Y, Z) = Tr{X R(X, Y )Z}. Cor2. For the Ricci tensors, we have Ric(Y, Z) = Ric(Y, Z) η(r(ξ, Y )Z) η(z) Ric(Y, ξ) + ( Y η)(z) div ξ ( Y ξη ) (Z), where div ξ = Tr{X X ξ}. Zbigniew Olszak Schouten-van Kampen affine connections - 9/21

10 The Riemannian case, and dim H = n 1 (6) Define the scalar curvatures by r = Tr g {(Y, Z) Rc(Y, Z)}, r = Tr g {(Y, Z) Rc(Y, Z)}. Cor3. For the scalar curvatures, we have r = r 2ε 0 Rc(ξ, ξ) + ε 0 (Div ξ) 2 Tr g {(Y, Z) ( Y ξη ) (Z). Zbigniew Olszak Schouten-van Kampen affine connections - 10/21

11 The Riemannian case, and dim H = n 1 (7) For a non-degenerate section (a 2-dimensional subspace) π = Span{X, Y }, X, Y T p M, p M, the sectional curvatures of and are defined by R(X, Y, Y, X ) K(π) = g(x, X )g(y, Y ) g 2 (X, Y ), K(π) = R(X, Y, Y, X ) g(x, X )g(y, Y ) g 2 (X, Y ). Cor4. For the sectional curvatures, we have 1 K(π) = K(π) + g(x, X )g(y, Y ) g 2 (X, Y ) ( η(x )R(X, Y, Y ξ) + η(y )R(Y, X, X, ξ) + ε 0 ( X η)(x )( Y η)(y ) ε 0 ( X η)(y )( Y η)(x ) ). Zbigniew Olszak Schouten-van Kampen affine connections - 11/21

12 Almost (para-)hermitian structures A unification of the Hermitian and para-hermitian notations. N - 2n-dimensional differentiable manifold, J, G - a (1, 1)-tensor field and a (pseudo-)riemannian metric on N and J 2 X = ε 1 X, G(JX, JY ) = ε 1 G(X, Y ), ε 1 = ±1. ε 1 = 1 (J, G) - an almost Hermitian structure on N, ε 1 = 1 (J, G) - an almost para-hermitian structure on N. N and (J, G) will be called almost (para)hermitian. Ω - the fundamental form defined by Ω(X, Y ) = G(X, JY ). This is a skew-symmetric (0, 2)-tensor field of maximal algebraic rank (an almost symplectic form) on M. Zbigniew Olszak Schouten-van Kampen affine connections - 12/21

13 Almost (para)contact metric structures (1) A unification of the contact and paracontact notations. M - a (2n + 1)-dimensional connected differentiable manifold, ϕ - a (1, 1)-tensor field, ξ - a vector field, η - a linear form and ϕ 2 X = ε 1 (X η(x )ξ), η(ξ) = 1, ε 1 = ±1. (ϕ, ξ, η) is an almost contact (resp., paracontact) structure when ε 1 = 1 (resp., ε 1 = 1). g - a (pseudo-)riemannian metric on M such that g(ϕx, ϕy ) = ε 1 (g(x, Y ) ε 0 η(x )η(y )), ε 0 = ±1. Consequences: ϕξ = 0, η ϕ = 0, η(x ) = ε 0 g(x, ξ), g(ξ, ξ) = ε 0. M and (ϕ, ξ, η, g) will be called almost (para)contact metric. Φ defined by Φ(X, Y ) = g(x, ϕy ) is the fundamental form. Zbigniew Olszak Schouten-van Kampen affine connections - 13/21

14 Almost (para)contact metric structures (2) An almost (para)contact metric manifold (structure) is called: (1) normal if the almost (para)complex structure J defined on M R by JX = ϕx + η(x ) t, J t = ε 1ξ is integrable, or equivalently, [ϕ, ϕ](x, Y ) 2ε 1 dη(x, Y )ξ = 0, [ϕ, ϕ] being the famous Nijehuis torsion tensor of ϕ; (2) (para)contact if Φ = dη; (3) K-(para)contact if it is (para)contact and ξ is a Killing vector field; (4) (para)sasakian if it is normal and (para)contact, or equivalently, ( X ϕ)y = ε 1 (g(x, Y )ξ ε 0 η(y )X ). Zbigniew Olszak Schouten-van Kampen affine connections - 14/21

15 SvK. aff. conn. adapted to almost (para)contact metric structures (1) M - an almost (para)contact metric manifold, dim M = 2n + 1, (ϕ, ξ, η, g) its almost (para)contact metric structure. H := ker η - the (para)contact distribution, dim H = 2n, V := span{ξ}, dim V = 1. Df3. The Schouten-van Kampen connection adapted to the pair (H, V) and associated the Levi-Civita connection arising from g will be called the Schouten-van Kampen connection adapted to the almost (para)contact metric structure (ϕ, ξ, η, g). Recall that it is given by the formula X Y = X Y η(y ) X ξ + ( X η)(y )ξ. Hint. The Schouten-van Kampen connection is different from the (generalized) Tanaka-Webster connections and another affine connections adapted to contact metric structures. Zbigniew Olszak Schouten-van Kampen affine connections - 15/21

16 SvK. aff. conn. adapted to almost (para)contact metric structures (2) Th7. With respect to the Schouten-van Kampen connection adapted to an almost (para)contact metric structure, ( X ϕ)y = ( X ϕ)y + η(y )ϕ X ξ ε 0 g(ϕ X ξ, Y )ξ. Th8. Let be the Schouten-van Kampen connection adatped to an almost (para)contact metric structure. The almost (para)contact metric structure is normal if and only if ( ϕx ϕ)ϕy + ε 1 ( X ϕ)y = 0 and B v (ϕx, ϕy ) = ε 1 B v (X, Y ). Zbigniew Olszak Schouten-van Kampen affine connections - 16/21

17 SvK. aff. conn. adapted to almost (para)contact metric structures (3) Th9. An almost (para)contact metric manifold is (para)contact if and only if the torsion T of the Schouten-van Kampen connection adapted to the almost (para)contact metric structure realizes the condition T v = 2Φ ξ. Th10. An almost (para)contact metric manifold is K-(para)contact if and only if the second fundamental form B associated to the Schouten-van Kampen connection adapted to the almost (para)contact metric structure realizes the condition B v = Φ ξ, or equivalently, B h = ε 0 ϕ η. Zbigniew Olszak Schouten-van Kampen affine connections - 17/21

18 SvK. aff. conn. adapted to almost (para)contact metric structures (4) Th11. Let be the Schouten-van Kampen connection adapted to an almost (para)contact metric structure. The following conditions are equivalent (a) the structure is (para)sasakian; (b) for the connection, it holds that ϕ = 0 and B h = ε 0 ϕ η; (c) for the connection, it holds that ϕ = 0 and B v = ξ Φ. Zbigniew Olszak Schouten-van Kampen affine connections - 18/21

19 SvK. aff. conn. adapted to almost (para)contact metric structures (5) When dim M = 3, an arbitrary almost (para)contact metric structure satisfies ( X ϕ)y = η(y )ϕ X ξ + ε 0 g(ϕ X ξ, Y )ξ. Therefore, we have: Cor5. For a 3-dimensional almost (para)contact metric structure, (a) ϕ = 0; (b) the structure is normal iff B v (ϕx, ϕy ) = ε 1 B v (X, Y ); (c) the structure is (para)sasakian iff B h = ε 0 ϕ η, or equivalently, B v = ξ Φ. Zbigniew Olszak Schouten-van Kampen affine connections - 19/21

20 Main references: (1) for Schouten-van Kampen connections J. Schouten and E. van Kampen, Zur Einbettungs- und Krümmungstheorie nichtholonomer Gebilde, Math. Ann. 103 (1930), S. Ianuş, Some almost product structures on manifolds with linear connection, Kōdai Math. Sem. Rep. 23 (1971), A. Bejancu and H. Faran, Foliations and geometric structures, Mathematics and Its Applications Vol. 580, Springer, Dordrecht, G. Pripoae, Connexions de Schouten et de Vrănceanu sur des f -variétés, Riv. Mat. Univ. Parma (4) 12 (1986), A.F. Solov ev, Second fundamental form of a distribution, Math. Notes 31 (1982), 71-75; transl. from Mat. Zametki 31 (1982), A.F. Solov ev, Curvature of a distribution, Math. Notes 35 (1984), 61-68, transl. from Mat. Zametki 35 (1984), A.F. Solov ev, Curvature of a hyperdistribution and contact metric manifolds, Math. Notes 38 (1985), ; transl. from Mat. Zametki 38 (1985), Zbigniew Olszak Schouten-van Kampen affine connections - 20/21

21 (2) for alm. (para-)hermitian and alm. (para)contact str. D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics Vol. 203, Birkhäuser, Boston, A. Bonome, R. Castro, E. Garcìa-Rìo and L. Hervella, Curvature of indefinite almost contact manifolds, J. Geom. 58 (1997), M. Brozos-Vázques, E. García-Río, P. Gilkey, S. Nikčević and R. Vázques-Lorenzo, The geometry of Walker manifolds, Synthesis Lectures on Mathematics and Statistics Vol. 5, Morgan & Claypool Publishers, San Rafael, V. Cruceanu, P. Fortuny and P.M. Gadea, A survey on paracomplex geometry, Rocky Mountain J. Math. 26 (1996), S. Ivanov and S. Zamkovoy, Parahermitian and paraquaternionic manifolds, Diff. Geom. Appl. 23 (2005), M.M. Tripathi, E. Kılıç, S.Y. Perktaş, and S. Keles, Indefinite almost paracontact metric manifolds, Int. J. Math., Math. Sci. Vol. 2010, Article ID , 19 p. Zbigniew Olszak Schouten-van Kampen affine connections - 21/21