By Kollar J., Lazarsfeld R., Morrison D. (eds.)
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Extra resources for Algebraic Geometry Santa Cruz 1995, Part 2
E. x2M Tx? M 6D ;. To be full is not really a restriction for submanifolds M of the Euclidean space Rn since we may always consider the smallest affine subspace that contains M . Contrary to that, there are many submanifolds of the pseudo-Euclidean space Rp;q that are contained in an affine subspace that is degenerate with respect to the inner product but not in a proper non-degenerate one. This is also the case if one restricts oneself to extrinsic symmetric spaces. As far as normality is concerned, Ferus proved in  that extrinsic symmetric spaces in the Euclidean space decompose into a product of an affine subspace and a normal extrinsic symmetric space.
L; Âl ; 0/. Then the bundle projection p W T N ! N is the simplest example of a special affine fibration over N . l; Â l /. We now want to construct special affine fibrations that correspond to quite general quadratic extensions in the same way as p W T N ! l; Â l ; 0/. l/ \ lC / and Âl0 is induced by Â l . 8. l; Â l / be a proper Z2 -equivariant Lie algebra. l; Â l /-module. g; Â; h ; i/. We assume in addition that at least one of the following two conditions is satisfied: (a) M is simply connected.
2/ D fŒH; X D 2Y; ŒH; Y D 2X; ŒX; Y D 2H g; lC D R H; l D spanfX; Y g pqr 0;n 2Cp q . Â l ; Âa /. Z; l/ al g. Z; l/ D a g. Z; l/ D al 6D 0g ! l; Â l ; a/0 ˛ 7 ! Œ˛; 0 (9) is a bijection. Let us now determine a suitable set Anl;Â l . Let . l; Â l ; a/0 6D ;. l/ D R Z. Hence, can be considered as a semisimple representation of the abelian Lie algebra l , which is determined by its weights. Moreover, we know from (9) that al D a0;1 or al D a0;2 . For D . l n 0/p and 0;pCq D . l n 0/q we define a representation ; of l on ap;q C ˚a p;q 0;pCq p;p 0;2q with R ˚R .