On convex hypersurfaces in a space form

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 + 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 - ^LY - 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 kcxay:> = - 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 + -Z - 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 X III III m in + + + L J in L J III x J III After doing the necessary calculations we obtain 2< <7yY,Z> = 2y- 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 - *" e ** e j i t i J i - E + M e e ti J j ı i +e, E CJ7 L3L-1e>- &* e j **. e t J J i i - E 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.. 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 - Le, L-1C7-L3e> + e ı e. ı ». J + 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 >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 pointCcKeyword: Konveks hiperyüzey = Convex hypersurfaces ; Riemann manifoldu = Riemann manifold