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By Tracy Kompelien

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3 Involutive distributions of codimension one in Kahler manifolds Let (M, g, J, ry) be a 2n-dimensional Kahler manifold with unit 1 form rj If the distribution A of r\ is involutive then the covanant derivative Vrj satisfies the equalities (10) Let now ( V, 5, J, 17) be a Hermitian vector space with unit 1 form 77 The unit vector corresponding to r\ and the nullity space of r\ are denoted by £ and A respectively Further fj denotes the 1 -form corresponding to the vector J£ and AO denotes the vector space of vectors perpendicular to £ and J£ We denote by C, the linear space of all tensors L over V of type (0 2) satisfying the properties ( 10) o) = L(y0,x0), Let {e,} x = l 2(n - 1) be an orthonormal basis for the space A0 and go be the restriction of the scalar product onto A0 Then we consider the trace of the tensor L on AO Here is the inverse matrix of INVOLUTIVE DISTRIBUTIONS OF COD1MENSION ONE M Further we associate the following scalars and 1 forms with the tensor L 6(X] = L ( £ , X ) - p f j ( X ) , 8*(X) = It follows from (11) that 9(x0)=L(t,x0), 0( J£) = 0 ( 0 = 0 , The subgroup of the unitary group U(n) preserving the structure (g, J, 77) is the group U(n) x 7(2) where 7(2) denotes the unit matrix of order 2 Taking into account the representations X = x0 + fj(X)J£ + r,(X)t, Y = y0 + ij(Y)Jt we introduce the following tensors (projection operators) o) - Li(X,Y) = L2(X,Y) = L3(X,Y) = L(Jx0,Jy0) 2 tr0i _ L(x0,yo)+L(Jx0,Jyo) )= tr0L -p'fj(X)fi(Y), L1(X,Y)=Pn(X)fj(Y) These tensors determine the following subspaces of C £t = {Le£\L = Ll}, i = l, ,7 For any tensor L £ £ we have L = LI + + L7 We note that the tensors g(xo,yo) fj(X)rj(Y) and rj(X)fj(Y) are the invariant under the action of U(n - 1) x 1(2) tensors in the space £ The action of U(n 1) x 7(2) on the space C\ @ £2 0 £3 reduces to the action of U(n - 1) Taking into account that the decomposition C\ ® £3 ® £3 is irreducible under the action of 17 (n — 1) we obtain Proposition 3 1 (Decomposition into basic classes) £ = £i® ®£ 7 , where the factors £,(« = !

N —o2 Let V be the Levi-Civita connection of the standard metric g (the canonical z z connection) in En Denoting by W — —j= and £ = —== from equalities (1) \E 44 G GANCHEV and V M1LOUSHEVA we obtain the following derivative formulas for M2 with respect to conformal parameters (u, v) ' where summation convention is assumed Let V be the induced connection on (M2 , g) and a - the second fundamental tensor of M2 We denote by XM2 the Lie algebra of all C°° vector fields on M2 The Riemanman curvature tensor R of V is given by the equalities R(X, Y)Z = VXVYZ - VYVXZ - V[x Y ] Z , fl(jr,y,z,^) (4) = X, Y, Z e A-, Y, z, u e ff(fl(jr,y)z,y), Taking into account the Gauss formula from (3) and (4) we calculate the Riemanman curvature K of M2 p2 I p2 K -n.

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