ON NORMALLY FLAT EINSTEIN SUBMANIFOLDS

The purpose of this paper is to study the second fundamental form of some submanifolds M in Euclidean spaces E" which have flat normal connection. As such, Theorem gives precise expressions for the (essentially 2) Weingarten maps of all 4-dimensional Emstem submanifolds in E6, which are specialized in Corollary 2 to the Rcciflat submanifolds. The main part ofthis paper deals with fiat submanifolds. In 1919, E. Caftan proved that every flat submanifold" of dimension < 3 in a Euclidean space is totally cylindrical. Moreover, he asserted without proof the existence of flat nontotally cylindrical submanifolds of dimension > 3 in Euclidean spaces We will comment on this assertion, and in this respect will prove, in Theorem 3, that every flat submanifold M" with flat normal connection in lg is totally cylindrical (for all possible dimensions n and rn).

A Riemannian manifold is Einstein if its Ricci tensor field is proportional (with a constant coefficient of proportionality) to the Riemannian metric.We recall that every space of constant sectional curvature is Einstein The converse statement is true also in 2 and 3 dimensions, as shown by J.A Schouten and D.J

Struik in 1921
FACT A (see lSl or [51 or [1]).If a Riemannian manifold M of dimension n (n _< 3) is Einstein, then it is a space of constant curvature.
T.Y.Thomas in 1936 and A. Fialkow in 1938 classified the Einstein hypersurfaces of the real space forms In particular, we have FACT B (see [9] or [6] or [10] and [1]).Let M be a hypersufface immersed in En+l, where n _> 3.If M is Einstein, then: (B. 1) the Riemannian scalar curvature, say 8, of M is constant and non-negative, (B.2) if 8 0, then M is locally Euclidean; (B.3) if 8 > 0, then every point of M is umbilical and M is locally a hypersphere S'.
Theorem of this paper determines all possible expressions of the second fundamental form of all Einstein 2-codimensional submanifolds with flat normal connection in E6, and in Corollary 2 we specify these expressions for all Ricci flat 2-codimensional submanifolds with flat normal connection in E. The proofs of these two results use the flatness of the normal connection and are based on the following well- known characterization of4-dimensional Einstein spaces by I.M. Singer and T.A. Thorpe.
FACT C (see [9] or [1]).Let M be a Riemannian 4-manifold.Then M is Einstein if and only if, for every m E M, for any 2-plane P at n, the sectional curvature of P is equal to the sectional curvature ofthe 2-plane P+/-perpendicular to P at n.
The method of the proof of Theorem inspires us to establish in Theorem 3 a relation between flatness and cylindricity.The importance ofthis relation will be justified in Fact D. 2. STATEMENTS OF THE MAIN RESULTS.THEOREM 1.Let M be a 4-manifold isometrically immersed with flat normal connection in E. Then M is Einstein if and only if for each point m E M: (1.1) either M is cylindrical at m; (1.2) or M is umbilical (non-geodesic) with respect to a normal direction N1 at m and cylindrical in another normal direction N2 perpendicular to N1 at m; (1.3) or with respect to a suitable onhonormal tangent frame of M at m and an orthonormal normal frame {N1, N2} at , the Weingarten operators AN1, AN, admit respectively one among the following matricial representations: where ab cd; a 0 0 0 where a is a non-zero real number, and N2 is cylindrical; (1.3.3)where 4-1, a, b, p, q are real numbers such that ab 0 and pq e(a b2 ); 0 -0 0 0 p 0 0 (1.3.4)Av, 0 0 0 Av,= 0 0 q 0 a 0 00'u 0 0 0 where a, b, c, d, p, q, u are real numbers such that a -0, and pq=d-, pu=c-, qu b-, and (b-cd bd (d-> o. With respect to case 1.3.1 of Theorem 1, we give in particular the following EXAMPLE AND REMARK 1.Let Ml(c) and M2(c) be two surfaces of constant Gauss curvature c in the Euclidean 3-space lg 3. Then (1) the Riemannian product M 4 M] (c) x M2(c) canonically isometrically immersed in lg is an Einstein 2-dimensional submanifold with flat normal connection.It is not a space of constant curvature and moreover it is not Ricci fiat, unless c 0.
(2) In particular, for c < 0, for instance M](c) and M2(c) both being a pseudo-sphere in E of the same pseudo-radius c, the Riemannian product manifold M is an Einstein submanifold with flat normal connection in E 6 which has strictly negative scalar curvature.Thus, in contrast to the fact that for 1-codimensional Einstein submanifolds in Euclidean spaces the scalar curvature s R+, there exists 2- codtmensional Einstein submanifolds with any given real number as scalar curvature.
COROLLARY 2. Let M be a 4-dimensional manifold isometrically immersed with flat normal connection in lg.
Then M is Ricci flat if and only if for each m E M; (2.1) either M is flat (hence cylindrical) at m; (2.2) or with respect to a suitable orthonormal tangent frame at rn and an orthonormal normal frame {Na, N2) at m, the Weingarten operators AN ,AN admit respectively one of the following matricial representations: 0 -0 0 0 p 0 0 (2.21) AN,= 0 0 -0 AN= 0 0 q 0 0 0 0 a 0 0 0 0 where pq a > 0.
Then M" is flat if and only if it is cylindrical.3. DEFINITIONS [3].
We consider a manifold M isometrically immersed with codimension N in the Euclidean space ]n+N.
We shall say that M is quasi-umbilical in the direction if the Weingarten tensor A of admits an eigenvalue A with multiplidty n I or n.
In particular: (i) if (ii) if 3.2.M is (totally) cylindrical [resp.quasi-umbilical] if, locally around each point, there exists an orthonormal normal frame field composed with cylindrical [resp.quasi-umbilical] directions.Now we prove our results. 4.1 PROOF OF THE THEOREM 1.
Let M be a 4-manifold isometrically immersed in the Euclidean 6-space Suppose that M is Einstein.Then by Fact C, for any m E M and any 2-plane P in T,M, its sectional curvature is the same as the sectional curvature of its orthogonal 2-plane px in T,,M.
To exploit this statement, we suppose moreover that the normal eormeetion of M in E is fiat.
To resolve ts system of 6 equations th 7 unos, let us first compute A1 Using the equations ( 1), ( 2), (3) d the equity b + c + d s (where s is the constt scg cuature of M) we find that A1 is a lution of the equation: where x is uo md H is the m cuature vector at m.Such m equation adts a solution A] a since: 4 < H,N] > s (A1 + A2 + Az + A4) A(A= + A + A) 0. To detene the uos A2, A3, A4, p2, ,, we discuss on the index of nomulliW r(m) of M at m, i.e., the r of the emm amre operator at m. Bause of the system (*), () e {0, 2, 4, }.
CASE l: =(m) O. Then M is flat (hence cci flat) at m.By the system (**), M is cyfinddc at m. CASE 2: (m) 2. Then we obtn the situation (l. .
In acrdmce th ch of the possibilities om Theorem md Coroll 2, we cm ngmct loc pettion of submfolds of codimeion 2 in Eo th flat no come,ion wch e, at a pil poim, Einge or pmicul cci flat.
With respe= to .2),we consider the case of dimension n 4.
sume h: E x E4 EN is a flat bilin c p md consider the mension of the vor space [Imh] generated by e age of h.We may suppose thout loss of generity that N dim[Imh].Sin the mension of the space of sec b fos on E is equ to 10, we cm re oselves to 0 N 10.Using tecques for the proof of Fa .l ),it is sy to demonstrate tt, if N 6 {7, 8, 9,10}, we cm reduce N so that N 6 {0,1, 2, 3, 4, 5, 6}.In the se par where E. C prov Fa .1 ), he showed so that for the ce N 6 {0,1, 2, 3, 4} the flatness implies e cyfinddci.Consequemly, the oy uo ces e: "N 5" md "N 6".In 1986 [7], exple of a 4-mbmold in E I wch is, at a picul poim, flat thout being finddc is constm.However, a ll justifition of Fact .2) is =ill lacing for the moment; in other words the method of resolution of the so-cl Gauss equation of a flat submfold in a Eucfidem space is still uo in mension n d in codimension N th N n + 1, even for the case of dimension n 4.
One first resolution for such a problem is ven in Theorem 3 for the picul ce of flat no come=ion.
Then the onhogo projon of on the onhogon supplement subspace W l of W in w is flat too.