Afin daldırmalar ve total jeodezik afin daldırmalar
Affine immersions and totaly geodesic affine immersions
- Tez No: 39833
- Danışmanlar: PROF.DR. ABDÜLKADİR ÖZDEĞER
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Afin immersionlar, Jeodezik, Affine immersions, Geodesic
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
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. İV
Özet (Çeviri)
AFFINE IMMERSIONS and TOTALY GEODESIC AFFINE IMMERSIONS SUMMARY 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 writev^ = -^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 <j) (called the recurrence form). Two affine connections V and V on a differentiable manifold M are said to be protectively equivalent if there is a 1-form 6 such that V^y = VXY + 9{X)Y + 9{Y)X for all vector fields X, Y, on M. It is known that two affine connections are protectively equivalent if and onl if they have the same family of curves as pregeodesics. The Weyl projective curvature tensor P of an affine connection V on a manifold M is defined by P(X, Y)Z = R(X, Y)Z - (L(X, Y) - L(Y, X))Z + L(Y, Z)X - L(X, Z)Y where L(X, Y) = -(n2 - l)“1 [nRic(X, Y) + Ric(F, X)] Ric being the Ricci tensor given by Ric(X,y) = İz{Z - R{Z,X)Y} An affine manifold (M, V) is said to be locally protectively flat if in a neighborhood of each point of M the connection V is protectively equiv alent to an affine connection V which is fiat, that is, the curvature tensor tensor R' of V is identically zero. VIFundamental equations of Gauss and Codazzi for the affine immersion / can be written as tanSt*, Y)Z = R(X, Y)Z + Aa(XjZ)Y - Aa{YZ)X norR(X, Y)Z = (Vxa)(Y, Z) - (Vyc*)(X, Z) tanE(X,F)e = (VYA)çX - (VXA)^Y norR(X, Y)Ç = a(A^X, Y) - a(X, A(Y) + R±(X, Y)Ç for arbitrary vector fields X,Y,Z on M and a normal vector field f, Where R, R, R1- are the curvature tensors of the connections V, V and V”1- respectively. In case (M, V) is a hypersurface of (M, V) the above equations reduce to t<m R(X,Y)Z = R(X,Y)Z + h(X,Z)SY - h{Y,Z)SX noiR(X, Y)Z = (Vxh)(Y, Z) + r{X)h(Y, Z) - (Vyh)(X, Z) - r(Y)h(X, Z) İkxR\X,Y)Z = -(VXS){Y) + r(X)SY + (VYS)(X) - r(Y)SX norR(X, Y)Ç = -h(X, SY) + h(SX, Y) + 2t(X)t{Y). The covariant derivatives Vxo;, Vx A are defined by (Vxa)(Y, Z) = Vxa(F, Z) - a(VxY, Z) - a(Y, VXZ) (Vx A)eF = VxAçY - A(VXY - A ± Y In the case where (M, V) is projectively fiat (with symmetric Ricci Ten sor), we have R(X, Y)Z = t(F, Z)X - y(X, Z)Y, where 7 is normalized Ricci tensor for (M, V). In this case, all the formulus above become simpler. Thus we have R{X, Y)Z = 7(F, Z)X - 7(X, Z)Y + Aa{YZ)X - Aa(x,z)Y (Vxa)(F,Z) = (Vya)(X,Z) ( Vx A)çY + 7(F, OX = (yYA)tX + j(X, QY R^X, Y )£ = a(X, AçY) - a(AçX, Y) Following[4], equiaffine structures are considered, and the following re sults are obtained: (1) If M is a_hypersurface of M, / : (M,V) -? (M, V) an affine immersion and (M, V, W) an equiaffine structure, then VxW = t(X)W. viiConsequently, (M, V, W) is an equiaffine sturucture, if and only if T = 0. (2) If (M, V, W) and (M, V, W) are equiaffine stuructures and if / : (M, V) - »? (M,V) is an affine immersion, then an associated transversal vector field £ can be chosen to be equiaffine. For a totaly geodesic affine immersion / : (M, V) - * (M, V) the following results are obtained. (a) For a totaly geodesic affine immersion, the fundamental formulas of Gauss and Codazzi become simpler and take the forms R{X,Y)Z = R(X,Y)Z) R(X, Y)t = -(yxA)çY + ( Vy A);X + R^X, Y)Ç (b) Assume that / : (M, V) - ? (M, V) is a totaly geodesic affine im mersion and (M, V) is an affine manifold of recurrent curvature, say V R = <j> ® R. Then (M, V) is (a) flat or (b) of recurrent curvature, more precisely VJ2 = <j> (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 = <f> ® R. Then we have AtR(X, Y)W = - (R(X, Y)A)zW - (Vwy A)ÇX + ÇVWXA)(Y + <KW)((VYA)tX-(VxA)tY) (v^x)(x,y)e =4>(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 ~ <f>(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. IX
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