ON HYPERSURFACES WITH TYPE NUMBER TWO IN SPACE FORMS

MARIAN HOTLOŚ

Abstract

1. Basic notations

Let (M, g), n ≥ 3, be a connected semi-Riemannian manifold of class C^{∞}. We denote by ∇, R, C, S and κ the Levi-Civita connection, the Riemann-Christoffel curvature tensor, the Weyl conformal curvature tensor, the Ricci tensor and the scalar curvature of (M, g), respectively. The Ricci operator S is defined by g(SX, Y) = S(X, Y), where X, Y ∈ Ξ(M), Ξ(M) being the Lie algebra of vector fields on M. Next, we define the endomorphisms R(X, Y), C(X, Y) and X ∧_{A} Y of Ξ(M) by

(X ∧_{A} Y)Z = A(Y, Z)X − A(X, Z)Y ,
R(X, Y)Z = [∇_{X}, ∇_{Y}]Z − ∇_{[X,Y]}Z ,
C(X, Y)Z = R(X, Y)Z − \frac{1}{n − 2} (X ∧_{g} SY + SX ∧_{g} Y

−\frac{κ}{n − 1} X ∧_{g} Y)Z ,

respectively, where A is a symmetric (0, 2)-tensor and X, Y, Z ∈ Ξ(M). The Riemann-Christoffel curvature tensor R, the Weyl conformal curvature tensor C and the (0, 4)-tensor G of (M, g) are defined by

R(X_{1}, X_{2}, X_{3}, X_{4}) = g(R(X_{1}, X_{2})X_{3}, X_{4}) ,
C(X_{1}, X_{2}, X_{3}, X_{4}) = g(C(X_{1}, X_{2})X_{3}, X_{4}) ,
G(X_{1}, X_{2}, X_{3}, X_{4}) = g((X_{1} ∧_{g} X_{2})X_{3}, X_{4}) ,

respectively. For a (0, k)-tensor field T, k ≥ 1, and a (0, 2)-tensor field A on (M, g) we define the tensors R · T and Q(A, T) by

(R · T)(X_{1}, …, X_{k}; X, Y) = −T (R(X, Y)X_{1}, X_{2}, …, X_{k})

− ⋯ − T (X_{1}, …, X_{k−1}, R(X, Y)X_{k}) ,

Q(A, T)(X_{1}, …, X_{k}; X, Y) = −T ((X ∧_{A} Y)X_{1}, X_{2}, …, X_{k})

− ⋯ − T (X_{1}, …, X_{k−1}, (X ∧_{A} Y)X_{k}) ,

respectively. Putting in the above formulas T = R, T = S or T = C, A = g or A = S, we obtain the tensors R · R, Q(g, R), Q(S, R), R · S, Q(g, S), R · C, Q(g, C) and Q(S, C). We define the following subsets of M: U_{R} = \{x ∈ M | R − \frac{κ}{(n−1)n} G ≠ 0 at x\}, U_{S} = \{x ∈ M | S − \frac{κ}{n} g ≠ 0 at x\}, U_{C} = {x ∈ M | C ≠ 0 at x} and U = U_{S} ∩ U_{C}. We note that U ⊂ U_{R}.

A semi-Riemannian manifold (M, g), n ≥ 3, is said to be pseudosymmetric if at every point of M the tensors R·R and Q(g, R) are linearly dependent. This is equivalent to

R · R = L_{R} Q(g, R)1  (1)

on U_{R}, where L_{R} is some function on U_{R}. The class of pseudosymmetric manifolds is an extension of the class of semisymmetric manifolds ( R · R = 0). Some geometrical considerations show that (1) is a more natural curvature condition than the condition of semisymmetry. For a presentation of facts related to this statement and other conditions of pseudosymmetry type we refer to a recent review paper [1].

2. Main resuts

Let (M, g), n = dim M ≥ 4, be a semi-Riemannian manifold satisfying at every point the following curvature condition of pseudosymmetry type: the tensors R · C and Q(S, C) are linearly dependent. This is equivalent on the set U consisting of all points of M at which Q(S, C) ≠ 0 to

R · C = L Q(S, C) , 2  (2)

where L is some function on U. In this paper, without loss of generality, we restrict our investigations to the set U_{L} ⊂ U defined by U_{L} = {x ∈ U | L ≠ 0 at x}.

Our aim is to investigate hypersurfaces in semi-Riemannian space of constant curvature N_{s}^{n+1}(c), n ≥ 4, with signature (s, n + 1 − s). Let M be a such hypersurface. We denote by g the metric tensor of M induced from the metric of the ambient space and by H the second fundamental tensor of (M, g). Let U_{H} be the subset of M consisting of all points at which the tensor H^{2} is not a linear combination of H and g. In a few earlier papers we considered hypersurfaces in semi-Euclidean spaces 𝔼_{s}^{n+1}, n ≥ 4, fulfilling (2). For instance, in [4] (see Theorem 4.1 and Theorem 4.2) it was shown that if M is a hypersurface in 𝔼_{s}^{n+1}, n ≥ 4, satisfying (2) then on U_{H} ∩ U_{L} we have: rank (S − \frac{κ}{n−1} g) = 1 and R · C = Q(S, C) , i.e. (2) with L = 1. On the other hand we have

Proposition 1 ([3], Proposition 4.3) Let M be a hypersurface in N_{s}^{n+1}(c) satisfying R·C = Q(S, C). If U_{H} ⊂ M is nonempty then the ambient space must be semi-Euclidean.

Therefore we investigate hypersurfaces M in N_{s}^{n+1}(c) with nonzero sectional curvature c satisfying (2). The main result is the following

Theorem 1 ([3], Theorem 4.1) Let M be a hypersurface in N_{s}^{n+1}(c), c ≠ 0, satisfying (2). Then at every point x ∈ U_{H} ∩ U_{L} ⊂ M we have

\frac{κ}{n − 1} = \frac{κ̃}{n + 1} , S − \frac{κ}{n} g = β w ⊗ w , β ∈ , w ∈ T_{x}^{∗}M

and

L = \frac{1}{n − 1} , R·R = \frac{κ}{n(n − 1)} Q(g, R) , R·R = Q(S, R)−\frac{(n − 2)κ}{n(n − 1)} Q(g, C) .

From Theorem 5.1 of [2] it follows: (1) holds at a point x ∈ U_{H} if and only if at this point rank H = 2, i.e. the type number of M at this point is equal to 2. We have the following

Proposition 2 ([3], Proposition 5.1) Let M be a pseudosymmetric hypersurface in N_{s}^{n+1}(c), n ≥ 4.

(i) At every point x ∈ U_{H} ⊂ M we have

(a) rank (H^{2} − tr(H) H) = 1 , or (b) rank (H^{2} − tr(H) H) = 2 , 3 (3)

(ii) If c ≠ 0 and at x ∈ U_{H} we have (3)(b) then (2) cannot be satisfied at this point. (iii) If c ≠ 0 and at a point x ∈ U_{H} we have (3)(a), i.e.

H_{ij}^{2} − tr(H)H_{ij} = \frac{1}{ρ} a_{i}a_{j} , ρ ∈

then the following relations are fulfilled at x: (2) with L = 1/(n − 1), \frac{κ}{n−1} = \frac{κ̃}{n+1} and

a^{k}a_{k} = 0 , a^{k} = g^{jk}a_{j} , S_{ij} − \frac{κ}{n} g_{ij} = \frac{ε}{ρ} a_{i}a_{j} ,

Basing on this result we construct two examples of hypersurfaces with type number two. The first one satisfies the equality R · C = \frac{1}{n−1} Q(S, C) and the second one does not.

References

[1] M. Belkhelfa, R. Deszcz, M. Głogowska, M. Hotloś, D. Kowalczyk, and L. Verstraelen, A review on pseudosymmetry type manifolds, Proc. Banach Center, in print.

[2] R. Deszcz, Pseudosymmetric hypersurfaces in spaces of constant curvature, Tensor, N.S. 58 (1997), 253–269.

[3] R. Deszcz and M. Hotloś, On hypersurfaces with type number two in space forms, to appear.

[4] R. Deszcz, M. Hotloś, and Z. S.entürk, Quasi-Einstein hypersurfaces in semi-Riemannian space forms, Colloq. Math. 89 (2001), 81–97.

Institute of Mathematics Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50–370 Wrocław POLAND e-mail: hotlos@im.pwr.wroc.pl