© Hindawi Publishing Corp. ON HYPERSURFACES IN A LOCALLY AFFINE RIEMANNIAN BANACH MANIFOLD II

We prove that an essential hypersurface of second order in an infinite dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature.


Introduction. Let
x) = er 2 , e = ±1, 0 ≠ r ∈ R, is called an essential hypersurface of the second order in the space M (see [2]).

Hypersurface of nonzero constant Riemannian curvature in a locally affine
Banach manifold. Let M be a locally affine Banach manifold and assume that 1 g is a strongly nonsingular metric on M, then the pair (M, where the Banach spaces E and F are the models of the manifolds M and N with respect to the charts c, and d, respectively. Furthermore, we have that Ψ (x) = x is the model of the pointx with respect to the chart d, z = Φ(x) is the model ofx with respect to the chart c, and i is the model of i with respect to the charts c and d. Then we have an inclusion of a hypersurface of a semi-Riemannian Banach space E.
In this case, (2.1) is called the local equation of the submanifold N ⊂ M with respect to the charts c and d. Also N will be a Riemannian submanifold of M with induced metric 2 g, which is defined by the rule for allx ∈ N andX 1 ,X 2 ∈ TxN, where Txi : TxN → TxM is the tangent mapping of i at the pointx ∈ N (see [1]).
Assume that 2 g is a strongly nonsingular metric on N. Also we have that M and N are Riemannian manifolds with free-torsion connections 1 Γ and 2 Γ , respectively, such that [3,4]). Let X 1 ,X 2 ∈ F be the models ofX 1 ,X 2 ∈ TxN with respect to the chart d on N. Then Y 1 = Di x (X 1 ) and Y 2 = Di x (X 2 ) are the models ofX 1 andX 2 with respect to the chart c on M.
In this case, the local equation of (2.2) takes the form

Theorem 2.1. A local hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semi-Riemannian Banach space is an essential hypersurface of second order.
Proof. Let N be a local hypersurface of constant curvature K 0 of the Banach type in the Riemannian manifold (M, 1 g) such that dim N > 2. We know that the first differential equation of the hypersurface N ⊂ M has the form (see [5] for allx ∈ N ⊂ M and allX ∈ T x N, and A x is the second fundamental form for the hypersurface N which is defined by the equality (see [5]) Taking into account that T x i ∈ T 1+0 0+1 (N) is a mixed tensor of type (1 + 0, 0 + 1) on the submanifold N (see [7]),ξ x ∈ T 1 0 (M), and (2.6), we conclude that A x is a symmetric tensor of type (0, 2) on N at the pointx ∈ N. Now let ξ : with respect to the charts c and d at the pointx. Then the local equations of equalities (2.5) take the form 1 g ξ x ,ξ x = e, 1 g Di x (X), ξ x = 0, (2.8) for all x ∈ Ψ (V ) ⊂ F and all X ∈ F . Furthermore, the integral condition for (2.4) takes the form Remark 2.2. In formula (2.9), there exists an alternation with respect to the underlined vectors without division by 2. This convention will be used henceforth.
Similarly, the second differential equation of the hypersurface N ⊂ M will be (see [5]) where H x ∈ L(F ; F). Also by using (2.6), we find that for all x = Ψ (x) ∈ Ψ (V ) ⊂ F and all X, Y ∈ F . Furthermore, the integral condition for (2.10) has the form (see [5] (2.14) Since N is a hypersurface of constant curvature, then (2.14) takes the form (see [2]) where K 0 ∈ R is a constant independent of the choice of the point, and is called the curvature of the hypersurface N. Then, we obtain Now we prove that A x is a weakly nonsingular form. Let X be a fixed vector and A x (X, Y ) = 0, for all Y ∈ F . Then, from (2.16) we obtain and all X, Z, S ∈ F . Since dim E > 2, then, for any S, we can choose Z which is not a multiple of X and thus 2 g x (X, S) = 0, for all S ∈ F . But 2 g x is nonsingular, hence, X = 0 and this proves that A x is a weakly nonsingular form. Now from (2.12) and (2.16), we obtain and then we have Taking into account that the metric tensor 2 g x is nonsingular, we obtain Since dim F > 2, then, for every X, Y ∈ F such that 2 g x (X, Y ) = 0, there exists a vector S ∈ F orthogonal to each X and H x (X) [2]. Using this fact in (2.23) and taking into account (2.12), we obtain A x (X, Y )·H x (S) = 0. By using the nonsingularity of the tensor A x , we conclude that A x (X, Y ) = 0. Since, for any pair of vectors X, Y ∈ F , 2 g x (X, Y ) = 0 implies that A x (X, Y ) = 0, then there exists a real number λ such that (see [2]) (2.24) Substituting (2.24) into (2.16), we obtain for all x = Ψ (x) ∈ Ψ (V ) ⊂ F and all X, Y , Z, S ∈ F . Taking into account the nonsingularity of 2 g x , we obtain λ 2 = K = K 0 /e. It is convenient to put K 0 = e/r 2 , where r is a nonzero real number and e = ±1, then we have λ = ±1/r . We find that in our case, it is convenient to take λ = −1/r . Substituting λ in (2.24), we obtain This last equation shows that the hypersurface N ⊂ M of constant nonzero Riemannian curvature will be locally an essential hypersurface of second order, and this completes the proof.