Bir uzay formun konveks hiperyüzeyleri üzerine
On convex hypersurfaces in a space form
- Tez No: 14130
- Danışmanlar: PROF.DR. ABDÜLKADİR ÖZDEĞER
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Konveks hiperyüzey, Riemann manifoldu, Convex hypersurfaces, Riemann manifold
- Yıl: 1990
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Hasanis [2] deki çalışmasında Cn+D -boyutlu Euclid uzayının konveks hi per yüzeyi er i için A logdetL-div X-v CTrL,logdetL5=CTrL32 - nCTrL2>-A ve A logdetL-div jj-V CTrL, logdetL} = III ** XXX II w = CTrL>CTrL S - n2 + B formüllerini ispatlamıştır Bilindiği gibi, sabit Riemann eğrili ki i bir Riemann mani fol duna bir uzay formu denir. Baikoussis ve Koufogiorgos, Hasanis' in elde ettiği bu formülleri bir uzay formun konveks hi per yüzeyi er i ne genel leştir mistir [43- İki bölümden oluşan bu çalışmanın birinci bölümünde Riemann geometrisine ait bazı temel tanım ve teoremlere yer ver i 1 mi s ti r. îkinci bölümde, önce bir uzay formun bir hiperyüzeyi- nin üçüncü esas forma göre Riemann konneksiyonu tanımlan mış ve bunun yardımıyla hi per yüzeyin birinci ve üçüncü esas forumlarına göre Riemann eğrilik tensör 1 eri arasın daki ilişki elde edilmiştir. Daha sonra, [2] de Ökl i di yen hiperyüzeyleri için elde edilen iki formülün bir uzay formun hiper yüzeyi erindeki karşılığı elde edilmiş ve bazı Özel hallere ait sonuçlar üzerinde durulmuştur. IV
Özet (Çeviri)
ON CONVEX HYPERSURFACES IN A SPACE FORM SUMMARY In [E] Hasanis considers a convex hypersurface M in a C n+1 D -di mensi onal Euclidean space and establishes the formulae A logdetL-div X-V C Tr L, 1 ogdetLD =C Tr LZ> 2 - nCTrA-A and -1 A logdetL-div u-Sf CTrL, logdetL} = III s XXT II ö CTrLDCTrL-1} - n2 + B. n+1 Moreover, for an oval old M in E, he obtains the in tegral inequalities f v CTrL,logdetLZ>dM > o and f V C Tr L_1, detL} dM < O M“ M ”.where the equality in either case holds, if and only if, M is a hypersphare. We recall that a Rismannian manifold of conasmt Riemannian curvature is called a space form. The above formulae of Hasanis have been generalized by Bakoussis and Koufogiorgos [4], to a convex hypersurface of a space form. This work consists of two parts. In the first part, some fundamental concepts of Riemannian geometry, used in this work, are given. Let / be an isometric immersion of the n-di mensi on al Riemannian manifod M into the C n+1 3 -di mensi onal Riemannian manifold M and let N be the unit vector field on M with respect to the immersion /. Weingarten map for a hypersurface M is defined by LY = v* N, where v* denotes the Riemannian connection of M and Y is a vector field on M.Denoting the metrics of M and M »respectivly, by <, > and <~ > w© have the Gauss equation v\,YZ> = <7VY + <LX,Y>N. X X This equation defines the Riemannian connection V on M. On the other hand, the curvature tensor field R of M which is defined by RCX.YDZ = V^VyZJ-^CV^-V^^ Z, is related to the curvature tensor R of M by Gauss cur vature equation RCX.YDZ = RCX.Y3Z - ^<LX,Z>LY - <LY,Z>LX^, whereas the Codazzi equation of M takes the form CVVLDY = VVCLY3 - Lv“Y. XX X The sectional curavature KCX^YD for the oriantation XAY determined by the tangent vectors X and Y at peM, is defined by <RCX,YDY-,X> kcxay:> =<X,X><Y,Y> - <X,Y>2 If M is a Riemannian manifold of constant sectional curvature c and M is an immersed hypersurface of M, then there is a relation between the sectional curvatures of M and M. Namely, kcxay:> = c + <LX,X><LY,Y>-<LX,Y>Z <X,X><Y,Y>-<X,Y> Let / be an isometric immersion of a smooth, com pact, connected and orientable n-di mensi onal Riemannian manifold M into a smooth, connected C n+1 5 -di mensi onal Riemannian manifold M of constant sectional curvature c. In the second part of this work, the Riemannian con- III r nection v”with respect to the third fundamental form <, > has been defined by «.- Ill ?* VIIll 2< v“y,z> = x<y,z> +y<x,z> -z<x,y> X III III m in +<rz,x],Y> +<rz,y],x> +<r x,yj,z> L J in L J III x J III After doing the necessary calculations we obtain 2< <7yY,Z> = 2<?yY + L'CVDY.Z) X III A A III Since this is valid for arbitrary Z, it follows that III _, vxy = vxY + L cvxl:>y- Following the rule used to define the curvature » III R of M, the curvature tensor R r III connection V is defined by the relation I III III III III tensor R of M, the curvature tensor R relative to the III III III III III III III RCX.YDZ = V”C VVZD- VVC VV2D- V fv,» Z. Using the last two equations, we obtain a relation between the curvature tensors R and R as 111 -1 -1 RCX.Y3Z = RCX.YZJZ+VyCL XC VyL^23 -7yCL CVXLDZ3 + +L~1^CVxL>VyZ-CVyLDVxZ+C,7xL3CL~1CVYL5ZD- CV L5CL-1CVXL3ZD-CV px ^ LZ>Z^. A straight -for ward calculation shows that III _. RCX.Y^Z = L 3RCX,Y3LZ. For any peM, we choose a field of orthonormal frames e,e,...,e with respect to <, > in a neighbourhood 12 n of p such that at p, e. Ci=l,2,...,nD are eigenvectors of L and that Ve = 0. If k.Ci=l,2nD are the eigen- values of L at p, then Le. = k e.. Thus, the sectional t It. curvature K of the 2-plane e.Ae. is found as ij x 3 K.. = c +kk. III After taking the inner product of RCX.Y^Y by X with respect to the first fundamental form and setting viiX=e., Y=e. in the expression so obtained, we get III F< RCe.,e)e - RCe,e.I)e,e > = u j j v J J i = E <v <^E L.-1CV LDe.D,e.>- *“ e ** e j i t i J i - E <V CrC7 LDL-1e.^,e> + M e e ti J j ı i +<E L_1C^7 LZ>e, E CJ7 L3L-1e>- &* e j **. e t J J i i - E <L_1C7 LDe,C7 LZ>L-1e.>.. e. j' e. j V.J V V Calculating all the terms in the above equation we find divCgrad TrLD - A ClogdetL!) + V CTrL.logdetL} = = P + Q, where Q = ”ECk“-k 5ZCk.k 5_1K. and P =E<L~1C7 LDe.CV L3L_1e>.. e. j e. j This is the generalization of the first formula, ob tained in [2], to a convex hypersurface of a space form. The vectors L eC = k. e 5 Ci=i.2nD define an orthonormal basis in the tangent space at peM with respect to the third fundamental form. We extend these vectors to the vector fields e in a neighbourhood of p i ”" III_ _ such that, v1 e = O and e. = L~ e at p. III Taking the inner product of RCX.Y^Y by X with respect to the third fundamental form and setting X=e., Y=e. in the expression so obtained, we get III_ __ _ _ _ _ r< RCe.eDe - RCe,e)e,e> =.. ». J J İ J J t I I I vliiIll _ _ = E ?{< ?- CL-1CV- LDe Z>,e > -.. ' e e j ı in v, j ı j III - < 7- CL_ZC7- LZ>Le 3,i > - e e. t j in J i - <C7- LZ>Le, L-1C7-L3e> + e ı e. ı ». J + <L_1C7- LDe,C7- LDLe > J- e. j e. t ' 1- j Calculating all the terms in the above equation we find div Cgr ad TrL~S - A CloctdetLZ) - III II III * - 7 CTrL_1,logdetL:> = P + Q, where and Q = ECk. - k.Z>zCk k 3_2K. i<j v J v J tJ P= £k.<L *C7- L3e,L C7- L3e> >0... v e. j e. j ii The last formula is the generalization of the second formula, obtained in [2], to a convex hypersurface of a space form. If the sectional curvature K of M Cfor all 2-planesD is nonnegative when e<0, or K>c when c>0, then the fol lowing integral inequalities are obtained from the for mulae which are the generalization of Hasani s * s formulae I ir 7TTCTrL,logdetL3dM > O and I 7 CTrL_1,detL3dM < O. II Equality holds, if and only if, /is totally umbilical Ce>03, or, either /is totally umbilical or L has two distinct constant eigenvalues at every pointCc<OD. Finally, some special cases are studied with the use of the above inequalities. ix
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