## Afin daldırmalar ve total jeodezik afin daldırmalar

 dc.contributor.advisor Özdeğer, Abdülkadir dc.contributor.author Demirbüker, Hakan dc.contributor.authorID 39833 dc.contributor.department Matematik Mühendisliği tr_TR dc.date.accessioned 2023-03-03T13:03:53Z dc.date.available 2023-03-03T13:03:53Z dc.date.issued 1994 dc.description Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1994 tr_TR dc.description.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. tr_TR dc.description.abstract 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. en_US dc.description.degree Yüksek Lisans tr_TR dc.identifier.uri http://hdl.handle.net/11527/22350 dc.language.iso tr dc.publisher Fen Bilimleri Enstitüsü tr_TR dc.rights Kurumsal arşive yüklenen tüm eserler telif hakkı ile korunmaktadır. Bunlar, bu kaynak üzerinden herhangi bir amaçla görüntülenebilir, ancak yazılı izin alınmadan herhangi bir biçimde yeniden oluşturulması veya dağıtılması yasaklanmıştır. tr_TR dc.rights All works uploaded to the institutional repository are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. en_US dc.subject Matematik tr_TR dc.subject Afin immersionlar tr_TR dc.subject Jeodezik tr_TR dc.subject Mathematics en_US dc.subject Affine immersions en_US dc.subject Geodesic en_US dc.title Afin daldırmalar ve total jeodezik afin daldırmalar dc.title.alternative Affine immersions and totaly geodesic affine immersions dc.type Tez tr_TR
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