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(2) : 24^ m-^- uMait^Ns^ (M ii ^ A) TO 48^10^ Remarks on the Euler-Poincare Characteristic of a Hypersurface. Izumi HASEGAWA Department of Mathematics, Sapporo Branch, Hokkaido University of Education. :^-)\\W: ;@?®0^< 7- • ^°7^^ I^^-^^^OL^. § 1. Preliminaries. 1. Generalized Gauss-Bonnet formula. Let M be a compact orientable Riemannian manifold of an even dimension n{=2m) throughout this paper. The Euler-Poincare characteristic X(M) of M is given by. (1) X{M)={2lCn)\ KndV, 'M. where Cn is the volume of the Euclidian unit n-sphere, Kn donetes the Lipschitz-Killing curvature of M. and dV is the volume element of M. This formula is called the generalized Gauss-. Bonnet formula ([2], [3], [4]). 2. The Gauss equation for a hypersurface. Let M be an (%+l)-dimensional Riemannian manifold covered by a system of coordinate neighborhoods {V ; x1} and g^i and R^vm, the metric tensor and the curvature tensor respetively. Let M be covered by a system, of coordinate neighborhoods {U ; ua} and gab and Rated, the metric tensor and curvature tensor of M. respectively. Let M be immersed in M and Xi=Xl{tta) be the local parametric expression of M-. Throughout this paper, Greek indices run over the range {1, 2, ... ,n+l} and Latin indices the range {1, 2, ...,%}.. If we put (2) B^=9aXi, 9a=9l9ua, then, the Riemannian metric of M induced from that of M. is given by. (3) ga^g^BiB1,, and the equations of Gauss are presented by (4) Rabcd == R/t^VwBaB'bBcB" — HacHf,d + HadHfsc,. where Hab are the components of the second fundamental tensor H and Hab=Hba.. § 2. Some results. Theorem 1. Let M be a hyper surf ace of a space of constant curvature c^O (resp.. c^sO). If the second curvature tensor is ahuays positive (resp. negative), then X(M) {resp.. (1).
(3) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 (—1)"IX(M)) is non-negative. Proof. Because M is a space of constant curvature,. (5) R^=-c{g^g^-g^g^}-. From (3), (4) and (5), we have (6) Kabcd= -c{gacgM-gadgbc) - {HaMbd-HadH{,c) •. We can choose the orthonormal coordinate system for a tangent space of M so that Hab= 0{a =/= b) at any point of M. With respect to this coodinate system,. [k, 0. (7) H=. ky.. 0 ~kn where ki, k^, ... , kn are the eigenvalues of H.. If a, b, c, d are different mutually, -Rabal)=C+ka,kb,. ^a6ac=0 and Rabcd=0.. If, in particular, H is positive (resp. negative) and c^O (resp. c^O), —Rabab^Q (reSp. —Kabab^O)-. In this case, (9) ( - !}mKn = ^-^- -in^- -^RW^... Rin^in^n ^ 0. Comparison with (1) completes the proof. Q.E.D. Corollary. Let M. be a hypersurface of a space of constant curvature. If the sectional curvature of M. is always non-negative {resp. non-negative), then %{M") {resp. (—l)mX(M)) is non-negative.. Similarly we have. Theorem 2. Let M be a hypersurface of a conformally flat space M. If the sectional c'tirvature of M is always non-negative {resp. nonpositive) and R^^BaB's == agab + 0Hab,. zuhere a and ^ are the funtcions on M, then 7.(M} (resp. (—l)mX(M)) is non-negative. Remark 1. If j3=0 in the Theorem 2, then M is an Einstein space. In this case M is the space of constant curvature consequently.. Remark 2. It is sufficient that a neighborhood of any point of M. is immersed in M in these theorems.. References [1] Eisenhart, L. P. (1949), Riemannian Geometry, Princeton Univ., Princeton. [2] Allendoerfer, C. B. (1940), The Euler number of a Riemann manifold, Amer. J. Math., Vol. 62, p. 243-248. [3] Allendoerfer, C. B. and Well, A. (1943), The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc., Vol. 53, p. 101-129. [4] Chern, S. S. (1944), A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math., Vol. 45, p. 747-752. [5] Hasegawa, I. (1971), On the sectional curvatures and the Euler-Poincare characteristic of a Riemannian manifold, J. of Fac. Sci. Hokkaido Univ., Vol. 22, p. 62-66. [6] Tomonaga, Y. (1972), Euer-Poincar6 characteristic and sectional curvature, Differential Geometry, in honor of K. Yano, Kinokuniya, Tokyo, p. 501-502.. (2).
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