Afin daldırmalar ve total jeodezik afin daldırmalar

Abstract

İki bölümden oluşan bu çalışmanın birinci bölümünde afin daldırmalar ve eş-afin yapılara ait bazı temel tanım ve teoremlere yer verilmiştir. ikinci bölümde, (M, V) afin manifoldunun (M, V) afin manifolduna bir total jeodezik afin daldırması gözönüne alınmış ve / : (M, V) - ». (M, V) total jeodezik afin daldırmasında (M, V) manifoldunun rekürant eğrilikli olması halinde, (M, V) nin rekürant eğrilikli veya düz olması gerektiğini ifade eden teoremin ispatı verilmiştir. Ayrıca, bu koşullara ilave olarak, f nin ombilik ve M nin boyutunun üç veya üçten daha büyük olması halinde, (M, V) manifoldunun bir yerel projektif düz uzay olduğu sonucu elde edilmiştir.

In this work, after having given the fundamental concepts concerning the affine immersions, totaly geodesic affine immersions and equiaffine stuructures, some properties related to them are studied. By an affine manifold, we mean a pair (M, V), where M is a (con nected) difFerentiable manifold and V an affine connection on M. Let (M, V) and (M, V) be (connected) difFerentiable manifolds with torsion-free affine connections V and V and of respective dimensions n and m. An immersion / : M -*? M is called affine immersion if around each point of M there is a field of transversal subspaces x - *. Nx : Tf{x)M = f*(Tx(M))+-Nx such that for vectors fields X and Y on M we have a decomposition V/.(X)/*aO = /*(Vxy) + a(X, Y) where a(X, Y) ? Nx at each point x.a is said to be the second funda mental form of the immersion f. NX(M) will be called the normal space (rather than the transversal subspace) at x G M, and the assignment x 6 M - ? NX(M) will be called the normal bundle and will be denoted simply by N(M). If £ : x - ? £x is a normal vector field, then we write where A$X ? TX{M) and Vx£iVx at each point, A being the shape tensor, A$ the shape operator for £ and V is the connection in the normal bundle. We identify M, locally, with the image f(M) and simplify the denota tions by dropping the sign of the differential /» of the immersion / from the formulas and write v^ = -^x+.v^ If M is a hypersurface of M, then the formulas above take the form VxY = VxY + h(X,Y)Ç V^ = -5(X) + r(X)e Where S is a tensor field of type (1,1) and r is a 1-form. We call 5 the shape operator and r, the transversal connection form for /. If a = 0 at a point x, we say that / is totaly geodesic at x. Ifa = 0 at any point of M, we say that / is totaly geodesic. / is said to be umbilical at x e M if there is a 1- form p on NX(M) such that Aç = p(£)I for any £ G NX(M), where I denotes the identity transformation. If / is umbilical at every point of M, we say that / is umbilical. An afBne manifold (M, V) is said to be of recurrent curvature if its curvature tensor R is non-zero and satisfies the condition S7R = (j> ® R. for certain 1- form ® R. Then (M, V) is (a) flat or (b) of recurrent curvature, more precisely VJ2 = (g) R, (f> being the pull-back of the recurrence form 4> onto M. (c) For a totaly geodesic affine immersion, we have (VWR)(X, Y)Z = (VwR)(X, Y)Z (VwR)(X,Y)t =(R(X,Y)A)tW + AiR{X,Y)W - {VwxA)0T + (V2wy A)sX + (v^xx, y)e (d) Let / : (M, V) - > (M, V) be a totaly geodesic affine immersion, where (M, V) is an affine manifold of recurrent curvature, say V R = ® R. Then we have AtR(X, Y)W = - (R(X, Y)A)zW - (Vwy A)ÇX + ÇVWXA)(Y + (W)R\x,Y)t In particular, when / is additionaly umbilicial, i.e. Aç = p(Ç)I then p(OR(X, Y)W = - (R(X, Y)p)(t)W - ((V^y )P)(0 ~ (W)(VyP)(0)X + ((Vtx )p)(t) - d>(W)(VxP)(0)Y. viii (e) A sufficient condition for vanishing of weyl projective curvature tensor P(X, Y)Z = R(X, Y)Z - (L(X, Y) - L(Y, X))Z + L{Y, Z)X - L(X, Z)Y is given by the following theorem: Let P be the Weyl projective curvature tensor of an affine manifold (M, V). n = dim M > 3, and x a point of M. If there exist (0, 2)-tensors B and C such that RX(X, Y)Z = B(X, Y)Z ~ C(Y, Z)X + C(X, Z)Y for any X, Y, Z ? TX(M), then Px = 0. (f) Morever, let / : (M, V) -> (M, V) be an affine immersion. Then, if / is a totaly geodesic and umbilical with A ^ 0 (M, V) will be locally projectively flat if (a) (M, V) is of recurrent curvature and n = dim M > 3 or (b) (M, V) is an affine locally symmetric space.