PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 165–177 DOI: 10.2298/PIM141025008S

CURVATURE PROPERTIES OF SOME CLASS OF HYPERSURFACES IN EUCLIDEAN SPACES

Katarzyna Sawicz

Dedicated to Professor Makoto Yawata on his seventy-second birthday

Abstract. We determine curvature properties of pseudosymmetry type of hypersurfaces in Euclidean spaces 𝔼^{n+1} , n ⩾ 5, having three distinct nonzero principal curvatures λ_{1}, λ_{2} and λ_{3} of multiplicity 1, p and n−p−1, respectively. For some hypersurfaces having this property the sum of λ_{1}, λ_{2} and λ_{3} is equal to the trace of the shape operator of M. We present an example of such hypersurface.

1. Introduction

Let H be the second fundamental tensor of a hypersurface M immersed isometrically in a semi-Riemannian space of constant curvature N_{s}^{n+1} (c), with signature (s, n + 1 − s), n ⩾ 4, where c = \frac{_{κ}}{n(n+1)} and κ̃ is the scalar curvature of the ambient space. For precise definitions of the symbols used we refer to Section 2 of this paper and Sections 2 and 3 of [16] (see also [3, 5, 14, 34, 50]). Let U_{H} ⊂ M be the set of all points at which the tensor H^{2} is not a linear combination of H and the metric tensor g of M. Curvature conditions of pseudosymmetry type on hypersurfaces M in N_{s}^{n+1} (c), n ⩾ 4, satisfying on U_{H} ⊂ M the equation

(1.1)  H^{3} = tr(H)H^{2} + ψH,

where ψ is some function on U_{H}, were investigated in several papers: [1,7,8,11,12, 15, 16, 22, 23, 25, 26, 34, 37]. For instance, the Cartan hypersurfaces satisfy (1.1) (see, e.g., [12, Theorem 4.3], [16, Example 5.1(iii)]). Examples of hypersurfaces in Euclidean spaces 𝔼^{n+1} , n ⩾ 5, as well as in semi-Euclidean spaces 𝔼_{s}^{n+1} , with signature (s, n+1−s), n ⩾ 5, satisfying (1.1) are given in [1] and [11], respectively. For further examples we refer to [15, 16, 23, 26, 27, 31, 35].

2010 Mathematics Subject Classification: Primary 53B20, 53B25; Secondary 53C25.

Key words and phrases: Tachibana tensor, pseudosymmetry type curvature condition, hypersurface, principal curvature.

Communicated by Michael Kunzinger.

Curvature conditions of pseudosymmetry type on hypersurfaces M in N_{s}^{n+1} (c), n ⩾ 4, satisfying on M ∖ U_{H} the equation

(1.2)  H^{2} = ψH + ρg,

for some functions ψ and ρ on this set, were investigated among others in [2,7,9,18, 27, 33, 35, 45, 47, 54]. Examples of hypersurfaces in spaces of constant curvature satisfying (1.2) are given among others in [27, 35, 43, 56, 57]. It is obvious that (1.1) is a special case of a more general equation

(1.3)  H^{3} = ϕH^{2} + ψH + ρg,

where ϕ, ψ and ρ are some functions on U_{H}. Hypersurfaces M in N_{s}^{n+1} (c), n ⩾ 4, satisfying (1.3) on U_{H} ⊂ M were investigated for instance in [6, 21, 50]. Here we investigate curvature conditions of pseudosymmetry type on hypersurfaces M in 𝔼_{s}^{n+1} , n ⩾ 5, satisfying (1.3) on U_{H}. We can also consider (1.3) with ϕ = tr(H) on U_{H}, i.e., the equation

(1.4)  H^{3} = tr(H)H^{2} + ψH + ρg,

where ψ and ρ are some functions on U_{H}. Hypersurfaces M in N_{s}^{n+1} (c), n ⩾ 4, satisfying (1.4) on U_{H} ⊂ M were investigated in [4,30,51–53]. In [30, Proposition 2.1] it was proved that for every hypersurface M in N_{s}^{5}(c) equation (1.4) reduces on U_{H} ⊂ M to (1.1). Evidently, ρ = 0 on U_{H}. The assumption that dimM = 4 is essential. In Section 5 we present an example of a hypersurface M in 𝔼^{n+1} , n ⩾ 5, having at every point three distinct principal curvatures λ_{1}, λ_{2} and λ_{3} of multiplicity 1, p and q, respectively, where n = 1 + p + q, satisfying (1.4) with nonzero function ρ. In [50, Proposition 4.1] it was shown that the tensors R·C, C·R and C · C of a hypersurface M in N_{s}^{n+1} (c), n ⩾ 4, satisfying (1.3) on U_{H} ⊂ M are expressed on this set by a linear combinations of the Tachibana tensors Q(g, R), Q(S, R), Q(S, G), Q(H, G) and Q(S, g ∧ H), and the tensors g ∧ Q(H, H^{2}) and H ∧ Q(g, H^{2}). In Section 3 we present these formulas in the case when M is a hypersurface in 𝔼_{s}^{n+1} , n ⩾ 4. Further, in the next section we present these formulas in the special case when M is a hypersurface in 𝔼^{n+1} , n ⩾ 5, and at every point of the set U_{H} of a hypersurface M there are three distinct principal curvatures of multiplicity 1, p and p, respectively, where n = 2p + 1. In Section 5 we present an example of such hypersurface.

2. Preliminaries

Throughout the paper all manifolds are assumed to be connected paracompact manifolds of class C^{∞} . Let (M, g) be an n-dimensional, n ⩾ 3, semi-Riemannian manifold and let ∇ be its Levi-Civita connection and Ξ(M) the Lie algebra of vector fields on M. We define on M the endomorphisms X ∧_{A} Y and R(X, Y) of Ξ(M) by

(X ∧_{A} Y)Z = A(Y, Z)X − A(X, Z)Y,
R(X, Y)Z = ∇_{X} ∇_{Y}Z − ∇_{Y}∇_{X} Z − ∇_{[X,Y]}Z,

where A is a symmetric (0, 2)-tensor on M and X, Y, Z ∈ Ξ(M). The Ricci tensor S, the Ricci operator S, the tensor S^{2} and the scalar curvature κ of (M, g) are defined by S(X, Y) = tr{Z → R(Z, X)Y }, g(SX, Y) = S(X, Y), S^{2}(X, Y) = S(SX, Y) and κ = trS, respectively. The endomorphism C(X, Y) we define by

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.

Further, we define the (0, 4)-tensor G, the Riemann–Christoffel curvature tensor R and the Weyl conformal curvature tensor C of (M, g) by

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

respectively, where X_{1}, X_{2}, ⋯∈ Ξ(M).

Let B(X, Y) be a skew-symmetric endomorphism of Ξ(M) and let B be a (0, 4)-tensor associated with B(X, Y) by

(2.1)  B(X_{1}, X_{2}, X_{3}, X_{4}) = g(B(X_{1}, X_{2})X_{3}, X_{4}).

The tensor B is said to be a generalized curvature tensor [44] if

B(X_{1}, X_{2}, X_{3}, X_{4}) + B(X_{2}, X_{3}, X_{1}, X_{4}) + B(X_{3}, X_{1}, X_{2}, X_{4}) = 0,
B(X_{1}, X_{2}, X_{3}, X_{4}) = B(X_{3}, X_{4}, X_{1}, X_{2}).

Let B(X, Y) be a skew-symmetric endomorphism of Ξ(M) and let B be the tensor defined by (2.1). We extend B(X, Y) to a derivation B(X, Y)· of the algebra of tensor fields on M, by assuming that it commutes with contractions and B(X, Y) · f = 0, for any smooth function f on M. Now for a (0, k)-tensor field T, k ⩾ 1, we can define the (0, k + 2)-tensor B · T by

(B · T)(X_{1}, …, X_{k}; X, Y) = (B(X, Y) · T)(X_{1}, …, X_{k}; X, Y)

= −T (B(X, Y)X_{1}, X_{2}, …, X_{k}) − ⋯− T (X_{1}, …, X_{k−1} , B(X, Y)X_{k}).

If A is a symmetric (0, 2)-tensor then we define the (0, k + 2)-tensor Q(A, T) by

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

In this manner we obtain the (0, 6)-tensors B ·B and Q(A, B). Setting in the above formulas B = R or B = C, T = R or T = C or T = S, A = g or A = S, we get the tensors R · R, R · C, C · R, C · C, R · S, C · S, Q(g, R), Q(S, R), Q(g, C) and Q(g, S). Let A be a symmetric (0, 2)-tensor and T a (0, p)-tensor, p ⩾ 2. According to [22], the tensor Q(A, T) is called the Tachibana tensor of A and T, or the Tachibana tensor for short. We also remark that in some papers, the (0, 6)-tensor Q(g, R) is called the Tachibana tensor (see, e.g., [39–41,46,55]). For symmetric (0, 2)-tensors E and F we define their Kulkarni–Nomizu product E ∧ F by

(E ∧ F)(X_{1}, X_{2}, X_{3}, X_{4}) = E(X_{1}, X_{4})F (X_{2}, X_{3}) + E(X_{2}, X_{3})F (X_{1}, X_{4})
− E(X_{1}, X_{3})F (X_{2}, X_{4}) − E(X_{2}, X_{4})F (X_{1}, X_{3}).

Clearly, the tensors R, C, G and E ∧ F are generalized curvature tensors. For a symmetric (0, 2)-tensor E we define the (0, 4)-tensor \overline{E} by \overline{E} = \frac{1}{2}E ∧ E. We have \overline{g} = G = \frac{1}{2}g ∧ g. We note that the Weyl tensor C can be presented in the form

(2.2)  C = R − \frac{1}{n − 2} g ∧ S + \frac{κ}{(n − 2)(n − 1)} G.

We also have (see, e.g., [15, Section 3])

(2.3)  Q(E, E ∧ F) = −Q(F, \overline{E}).

Now (2.2) and (2.3) yield Q(g, C) = Q(g, R) + (1/(n − 2))Q(S, G). For a symmetric (0, 2)-tensor E and a (0, k)-tensor T k ⩾ 2, we define their Kulkarni–Nomizu product E ∧ T by [12]

(E ∧ T)(X_{1}, X_{2}, X_{3}, X_{4}; Y_{3}, …, Y_{k})
= E(X_{1}, X_{4})T (X_{2}, X_{3}, Y_{3}, …, Y_{k}) + E(X_{2}, X_{3})T (X_{1}, X_{4}, Y_{3}, …, Y_{k})
− E(X_{1}, X_{3})T (X_{2}, X_{4}, Y_{3}, …, Y_{k}) − E(X_{2}, X_{4})T (X_{1}, X_{3}, Y_{3}, …, Y_{k}).

Using the above definitions we can prove

Lemma 2.1. [11,12] Let E_{1}, E_{2} and F be symmetric (0, 2)-tensors at a point x of a semi-Riemannian manifold (M, g), n ⩾ 3. Then at x we have

E_{1} ∧ Q(E_{2}, F) + E_{2} ∧ Q(E_{1}, F) = −Q(F, E_{1} ∧ E_{2}).

If E = E_{1} = E_{2}, then

(2.4)  E ∧ Q(E, F) = −Q(F, \overline{E}).

3. Hypersurfaces in semi-Euclidean spaces

Let M, n ⩾ 3, be a connected hypersurface isometrically immersed in a semi-Riemannian manifold (N, g^{N}). We denote by g the metric tensor induced on M from g^{N}. Further, we denote by ∇ and ∇^{N} the Levi-Civita connections corresponding to the metric tensors g and g^{N}, respectively. Let ξ be a local unit normal vector field on M in N and let ε = g^{N} (ξ, ξ) = ±1. We can write the Gauss formula and the Weingarten formula of (M, g) in (N, g^{N}) in the form: ∇_{X}^{N} Y = ∇_{X} Y + εH(X, Y)ξ and ∇_{X}^{N} ξ = −AX, respectively, where X, Y are vector fields tangent to M, H is the second fundamental tensor of (M, g) in (N, g^{N}), A is the shape operator and H^{k}(X, Y) = g(A^{k}X, Y), k ⩾ 1, H^{1} = H and A^{1} = A. We denote by R and R^{N} the Riemann–Christoffel curvature tensors of (M, g) and (N, g^{N}), respectively. Let x^{r} = x^{r}(y^{k}) be the local parametric expression of (M, g) in (N, g^{N}), where y^{k} and x^{r} are local coordinates of M and N, respectively, and h, i, j, k ∈ {1, 2, …, n} and p, r, t, u ∈ {1, 2, …, n + 1}. The Gauss equation of (M, g) in (N, g^{N}) has the form

(3.1)  R_{hijk} = R_{prtu}^{N} B_{h}^{p}B_{i}^{r}B_{j}^{t}B_{k}^{u} + ε(H_{hk} H_{ij} − H_{hj} H_{ik}), B_{k}^{r} = \frac{∂x^{r}}{∂y^{k}} ,

where R_{prtu}^{N} , R_{hijk} and H_{hk} are the local components of the tensors R^{N}, R and H, respectively. If (N, g^{N}) is a conformally flat space then we have [23, Section 4]

C_{hijk} = μG_{hijk} + ε\overline{H}_{hijk} + \frac{ε}{n − 2} (g ∧ (H^{2} − tr(H)H))_{hijk} ,
(3.2)  μ = \frac{1}{(n − 2)(n − 1)} (κ − 2S̃_{rt} B_{h}^{r}B_{k}^{t}g^{hk} + κ̃),

where S̃_{rt} are the local components of the Ricci tensor S̃ of the ambient space, G_{hijk} are the local components of the tensor G and κ̃ and κ are the scalar curvatures of (N, g^{N}) and (M, g), respectively.

Let now M be a hypersurface in 𝔼_{s}^{n+1} , n ⩾ 4. Clearly, (3.1) and (3.2) read

(3.3)  R_{hijk} = ε\overline{H}_{hijk} , μ = \frac{κ}{(n − 2)(n − 1)} ,

respectively. Contracting (3.3) with g^{ij} and g^{kh} we obtain

(3.4)  S_{hk} = ε(tr(H)H_{hk} − H_{hk}^{2} ), κ = ε((tr(H))^{2} − tr(H^{2})),

respectively, where tr(H) = g^{hk} H_{hk}, tr(H^{2}) = g^{hk} H_{hk}^{2} and S_{hk} are the local components of the Ricci tensor S of M. We recall that on every hypersurface M in 𝔼_{s}^{n+1} , n ⩾ 3, we have the following identity R · R = Q(S, R) [32]. We prove now that on M in 𝔼_{s}^{n+1} , n ⩾ 3, we also have

Proposition 3.1. On every hypersurface M in 𝔼_{s}^{n+1} , n ⩾ 3, the following identities are satisfied

(3.5)  g ∧ Q(H, H^{2}) = εg ∧ Q(S, H),
(3.6)  H ∧ Q(g, H^{2}) = ε tr(H)Q(g, R) − εH ∧ Q(g, S).

Proof. From (3.4) we have

(3.7)  H^{2} = tr(H)H − εS,

and this yields

g ∧ Q(H, H^{2}) = g ∧ Q(H, tr(H)H − εS) = εg ∧ Q(S, H).

Thus (3.5) is proved. Further, using (2.4), (3.3) and (3.7) we obtain

H ∧ Q(g, H^{2}) = tr(H)H ∧ Q(g, H) − εH ∧ Q(g, S)
= − tr(H)H ∧ Q(H, g) − εH ∧ Q(g, S)
= tr(H)Q(g, \overline{H}) − εH ∧ Q(g, S)
= ε tr(H)Q(g, R) − εH ∧ Q(g, S).

Our proposition is thus proved. □

Let now M be a hypersurface in 𝔼_{s}^{n+1} , n ⩾ 4, satisfying (1.3) on U_{H} ⊂ M. We set (cf. [50, eq. (34)])

(3.8)

β_{1} = ε(ϕ − tr(H)),
β_{2} = −\frac{ε}{n − 2} (ϕ(2 tr(H) − ϕ) − (tr(H))^{2} − ψ − (n − 2)εμ),
β_{3} = εμ tr(H) + \frac{1}{n − 2} (ψ(2 tr(H) − ϕ) + (n − 3)ρ),
β_{4} = β_{3} − εβ_{2} tr(H),
β_{5} = \frac{κ}{n − 1} + εψ + β_{1} tr(H),
β_{6} = β_{2},

where the functions ϕ, ψ and ρ are defined by (1.3).

Proposition 3.2. If M is a hypersurface in 𝔼_{s}^{n+1} , n ⩾ 4, satisfying (1.3) on U_{H} ⊂ M, for some functions ϕ, ψ and ρ, then the following conditions are satisfied on this set

(3.9)  S^{2} = γ_{2}S + γ_{1}H + γ_{0}g,
(3.10)  (n − 2)R · C = (n − 2)Q(S, R) + ρQ(H, G) − β_{1}g ∧ Q(H, S),
(3.11)  (n − 2)C · R = ( \frac{κ}{n − 1} + εψ + β_{1} tr(H)) Q(g, R)
+ (n − 3)Q(S, R) − β_{1}H ∧ Q(g, S),
(3.12)  (n − 2)C · C = β_{1}Q(S, g ∧ H) + β_{4}Q(H, G)
+ (n − 3)Q(S, R) + β_{5}Q(g, R) + β_{2}Q(S, G),
γ_{0} = ρ(ϕ − 2 tr(H)),
(3.13)  γ_{1} = ψ(ϕ − 2 tr(H)) + ρ + tr(H)(ϕ^{2} + ψ + (tr(H))^{2}),
γ_{2} = −(ϕ^{2} + ψ + tr(H)(tr(H) − 2ϕ)).

Proof. We denote by S_{hk}^{2} the local components of the tensor S^{2}. Evidently, we have

S_{hk}^{2} = g^{ij} S_{hi} S_{kj} = H_{hk}^{4} − 2 tr(H)H_{hk}^{3} + (tr(H))^{2}H_{hk}^{2} .

Applying in this (1.3) we obtain

S^{2} = (ϕ^{2} + ψ + tr(H)(tr(H) − 2ϕ))H^{2}
+ ρ(ϕ − 2 tr(H))g + (ψ(ϕ − 2 tr(H)) + ρ)H.

The last relation, by making use of (3.7) and (3.13), turns into (3.9). Further, we also have on U_{H} (cf. [50, Proposition 4.1]): (3.12) and

(3.14) (n − 2)R · C = (n − 2)Q(S, R) + ρQ(H, G) + (ϕ − tr(H))g ∧ Q(H, H^{2}),
(3.15) (n − 2)C · R = ( \frac{κ}{n − 1} + εψ) Q(g, R) + (n − 3)Q(S, R)
+ (ϕ − tr(H))H ∧ Q(g, H^{2}),

where β_{1}, …, β_{5} are defined by (3.8). Now (3.10) and (3.11) are an immediate consequence of (3.5), (3.6), (3.8), (3.14) and (3.15). □

4. Hypersurfaces with three principal curvatures

In this section we consider hypersurfaces M in 𝔼^{n+1} , n ⩾ 5, having at every point of the set U_{H} ⊂ M three distinct principal curvatures λ_{1}, λ_{2} and λ_{3}. First we note that from (1.3) it follows that

(4.1)  ϕ = λ_{1} + λ_{2} + λ_{3}, ψ = −(λ_{1}λ_{2} + λ_{1}λ_{3} + λ_{2}λ_{3}), ρ = λ_{1}λ_{2}λ_{3}.

Moreover we assume that λ_{1}, λ_{2} and λ_{3} are of multiplicity 1, p and p, respectively. Evidently, n = 2p + 1. Further, (3.3), (3.4) and (3.8) lead to

tr(H) = λ_{1} + p(λ_{2} + λ_{3}), tr(H^{2}) = λ_{1}^{2} + p(λ_{2}^{2} + λ_{3}^{2}),

β_{1} = −\frac{n − 3}{2} (λ_{2} + λ_{3}), β_{2} = \frac{1}{n − 3} β_{1}^{2} = ψ + \frac{κ}{n − 1}

β_{3} = \frac{1}{n − 2} ( tr(H)β_{2} − ψβ_{1} + (n − 3)ρ),

β_{4} = −\frac{1}{n − 2} ((n − 3) tr(H)β_{2} + ψβ_{1} − (n − 3)ρ), β_{5} = β_{2} + tr(H)β_{1}. (4.2)

Using now (3.9) and (4.1) we find

γ_{0} = −λ_{1}λ_{2}λ_{3}(λ_{1} + (2p − 1)(λ_{2} + λ_{3})),
γ_{1} = p(p − 1)^{2}(λ_{2}^{3} + λ_{3}^{3}) + p(p − 1)(λ_{2}^{2} + λ_{3}^{2})λ_{1}
+ (3p^{2}(p − 2) + 4p − 1)λ_{2}λ_{3}(λ_{2} + λ_{3}) + (2p^{2} − 2p + 1)λ_{1}λ_{2}λ_{3},

γ_{2} = −(p − 1)(λ_{2}^{2} + λ_{3}^{2}) − (p − 2)λ_{1}(λ_{2} + λ_{3}) − (2p − 3)λ_{2}λ_{3}. (4.3)

From (3.4) and (4.2) it follows immediately that the eigenvalues ρ_{1}, ρ_{2} and ρ_{3} of the Ricci tensor S of M are expressed on U_{H} trought the following relations

ρ_{1} = λ_{1}(tr(H) − λ_{1}) = pλ_{1}(λ_{2} + λ_{3}),
ρ_{2} = λ_{2}(tr(H) − λ_{2}) = λ_{2}(λ_{1} + (p − 1)λ_{2} + pλ_{3}),
(4.4)  ρ_{3} = λ_{3}(tr(H) − λ_{3}) = λ_{3}(λ_{1} + pλ_{2} + (p − 1)λ_{3}).

Now (4.4) yields

(4.5) (ρ_{1} − ρ_{2})(ρ_{1} − ρ_{3})(ρ_{2} − ρ_{3}) = (λ_{1} − λ_{2})(λ_{1} − λ_{3})(λ_{2} − λ_{3})((p − 1)λ_{2} + pλ_{3})

(pλ_{2} + (p − 1)λ_{3})(λ_{1} + (p − 1)(λ_{2} + λ_{3})).

Proposition 4.1. Let M be a hypersurface in 𝔼^{n+1} , n = 2p + 1 ⩾ 5, having at every point of U_{H} ⊂ M three distinct principal curvatures λ_{1}, λ_{2} and λ_{3} of multiplicity 1, p and p, respectively. We have

(i) The Ricci tensor S of M has at a point x ∈ U_{H} three distinct eigenvalues ρ_{1}, ρ_{2} and ρ_{3} if and only if at this point we have

((p − 1)λ_{2} + pλ_{3})(pλ_{2} + (p − 1)λ_{3})(λ_{1} + (p − 1)(λ_{2} + λ_{3})) ≠ 0.

(ii) If the Ricci tensor S of M has at a point x ∈ U_{H} three distinct eigenvalues ρ_{1}, ρ_{2} and ρ_{3}, then γ_{1}, defined by (3.13), is nonzero at this point, and in a consequence H = γ_{1}^{−1} (S^{2} − γ_{2}S − γ_{0}g).

Proof. (i) follows immediately from (4.5).

(ii) Suppose that γ_{1} = 0 at x. Then from (3.9) it follows that S has at x only two distinct eigenvalues, a contradiction. □

The above results, together with (3.3), Lemma 2.1 and Proposition 3.2, imply

Theorem 4.1. Let M be a hypersurface in 𝔼^{n+1} , n = 2p + 1 ⩾ 5, having at every point of U_{H} ⊂ M three distinct principal curvatures λ_{1}, λ_{2} and λ_{3} of multiplicity 1, p and p, respectively. Let U ⊂ U_{H} be the set of all points at which Ricci tensor S of M has three distinct eigenvalues ρ_{1}, ρ_{2} and ρ_{3}. Then on this set we have

(4.6)  R = \frac{1}{2}γ_{1}^{−2} (S^{2} − γ_{2}S − γ_{0}g) ∧ (S^{2} − γ_{2}S − γ_{0}g),
(n − 2)R · C = (n − 2)Q(S, R) + \frac{ρ}{γ_{1}} Q(S^{2}, G)

+ ( β_{1} − \frac{ργ_{2}}{γ_{1}} ) Q(S, G) + \frac{β_{1}}{γ_{1}} g ∧ Q(S, S^{2}), (4.7)

(n − 2)C · R = (n − 3)Q(S, R) + ( \frac{κ}{n − 1} + εψ + β_{1} tr(H)) Q(g, R)

+ \frac{β_{1}γ_{0}}{γ_{1}} Q(S, G) − \frac{β_{1}γ_{2}}{γ_{1}} Q( g, \frac{1}{2}S ∧ S) − \frac{β_{1}}{γ_{1}} S^{2} ∧ Q(g, S), (4.8)

(n − 2)C · C = (n − 3)Q(S, R) + β_{5}Q(g, R)

+ ( β_{2} − \frac{2β_{1}γ_{0} + β_{4}γ_{2}}{γ_{1}} ) Q(S, G)

+ \frac{β_{1}γ_{2}}{γ_{1}} Q( g, \frac{1}{2}S ∧ S) + \frac{β_{1}}{γ_{1}} Q(S, g ∧ S^{2}) + \frac{β_{4}}{γ_{1}} Q(S^{2}, G). (4.9)

Remark 4.1. Let M be the hypersurface considered in Theorem 4.1. By making use of (4.6) we state that the curvature tensor R of M is expressed on U_{H} ⊂ M by a linear combination of the Tachibana tensors:

G = \frac{1}{2}g ∧ g, g ∧ S, g ∧ S^{2}, S ∧ S^{2}, \overline{S} = \frac{1}{2}S ∧ S, \overline{S}^{2} = \frac{1}{2}S^{2} ∧ S^{2}.

Results on hypersurfaces in N_{s}^{n+1} (c), n ⩾ 4, with the curvature tensor R having the above property are given in [21] and [52]. Hypersurfaces in N_{s}^{n+1} (c), n ⩾ 4, with the curvature tensor R which is expressed by a linear combination of the tensors g ∧ g, g ∧ S and S ∧ S were investigated in [35]. For instance, the Clifford torus S^{p}(\sqrt{p/n}) × S^{n−p} (\sqrt{(n − p)/n} ), 2 ⩽ p ⩽ n − 2, n ≠ 2p, has this property [35, Corollary 3.1]. We also mention that semi-Riemannian manifolds with the curvature tensor R expressed by a linear combination of the tensors g∧g, g∧S and S ∧S were introduced and investigated in [10]. For further results on this class of manifolds we refer to [13, 17, 19, 20, 24, 27, 29, 31, 36, 42].

5. Example

Example 5.1. (i)(cf. [48, Section 2], [49, Section 2]) Let α_{1} = α_{1}(t) and α_{2} = α_{2}(t) be positive smooth functions defined on an interval I = (0; t_{0}) ⊂ ℝ, t_{0} > 0, such that α′_{1} ≠ 0 and α′_{2} ≠ 0 for every t ∈ I, where α′_{1} = \frac{dα_{1}}{dt} and α′_{2} = \frac{dα_{2}}{dt} . Let x = x(t, u^{1}, …, u^{p}, v^{1}, …, v^{q}) be a parametric expression of a subset M of an (n + 1)-dimensional Euclidean space 𝔼^{n+1} , n = p + q + 1, p ⩾ 2, q ⩾ 2, defined by

x = α_{1}F_{1} + α_{2}F_{2}, (5.1)

F_{1} = (cosu^{1}, sinu^{1} cosu^{1}, …, sinu^{1} …sinu^{p−1} cosu^{p}, sinu^{1} …sinu^{p}, 0, …, 0),
F_{2} = (0, …, 0, cosv^{1}, sinv^{1} cosv^{1}, …, sinv^{1} …sinv^{q−1} cosv^{q}, sinv^{1} …sinv^{q}),

where u^{1}, …, u^{p}, v^{1}, …, v^{q} ∈ (0, \frac{π}{2} ) and 0 occurs (q + 1)-and (p + 1)-times, respectively. We set

(5.2)  ξ = β( − α′_{2}F_{1} + α′_{1}F_{2}), β^{−1} = \sqrt{α′_{1} ^{2} + α′_{2} ^{2}} .

Further, we have ⟨F_{1}, F_{1}⟩ = ⟨F_{2}, F_{2}⟩ = ⟨ξ, ξ⟩ = 1, where ⟨·, ·⟩ denotes the standard scalar product of 𝔼^{n+1} . Differentiating (5.1) we obtain

x′_{t} = \frac{∂x}{∂t} = α′_{1}F_{1} + α′_{2}F_{2},

x′_{k} = \frac{∂x}{∂u^{k}} = α_{1} \frac{∂F_{1}}{∂u^{k}} = α_{1}F′_{1k}, x′_{l} = \frac{∂x}{∂v^{l}} = α_{2} \frac{∂F_{2}}{∂v^{l}} = α_{2}F′_{2l}, (5.3)

where k ∈ {1, …, p} and l ∈ {p + 1, …, p + q}. Using (5.2) and (5.3) we can easy check that

(5.4)  ⟨ξ, x′_{t}⟩ = ⟨ξ, x′_{k}⟩ = ⟨ξ, x′_{l}⟩ = 0.

We assume that at x we have

μ_{0}x′_{t} + μ_{1}x′_{1} + ⋯+ μ_{p}x′_{p} + μ_{p+1} x′_{p+1} + ⋯+ μ_{p+q} x′_{p+q} = 0,

where μ_{0}, …, μ_{p+q} ∈ ℝ. The last relation, by (5.3), turns into

α_{1}( \frac{α′_{1}μ_{0}}{α_{1}} F_{1} + μ_{1}F′_{11} + ⋯+ μ_{p}F′_{1p})

+ α_{2}( \frac{α′_{2}μ_{0}}{α_{2}} F_{2} + μ_{p+1} F′_{2p+1} + ⋯+ μ_{p+q} F′_{2p+q} ) = 0.

This and the definitions of F_{1} and F_{2} lead to

\frac{α′_{1}μ_{0}}{α_{1}} F_{1} + μ_{1}F′_{11} + ⋯+ μ_{p}F′_{1p} = 0,

\frac{α′_{2}μ_{0}}{α_{2}} F_{2} + μ_{p+1} F′_{2p+1} + ⋯+ μ_{p+q} F′_{2p+q} = 0.

Since the vectors F_{1}, F′_{11}, …, F′_{1p}, resp. F_{2}, F′_{2p+1} , …, F′_{2p+q} are linearly independent vectors (see, e.g., [38, Example 2, pp.329–331]) at x we have

\frac{α′_{1}μ_{0}}{α_{1}} = μ_{1} = ⋯= μ_{p} = 0, \frac{α′_{2}μ_{0}}{α_{2}} = μ_{p+1} = ⋯= μ_{p+q} = 0.

Thus the vectors x′_{t}, x′_{1}, …, x′_{p}, x′_{p+1} , …, x′_{p+q} are linearly independent at every point of M. Therefore we can state that M is immersed isometrically in 𝔼^{n+1} . In addition, from (5.4) it follows that ξ is the unit normal vector field of M. Further, differentiating (5.2) we obtain

ξ′_{t} = \frac{∂ξ}{∂t} = −(α′_{2}β)′F_{1} + (α′_{1}β)′F_{2},

ξ′_{k} = \frac{∂ξ}{∂u^{k}} = −α′_{2}βF_{1k}, ξ′_{l} = \frac{∂ξ}{∂v^{l}} = α′_{1}βF_{2l}, (5.5)

where α′′_{1}= \frac{dα′_{1}}{dt} and α′′_{2}= \frac{dα′_{2}}{dt} . From (5.3) and (5.5) we obtain the Weingarten formula for M

ξ′_{t} = (α′′_{1}α′_{2} − α′_{1}α′′_{2})β^{3}x′_{t} = \frac{α′′_{1}α′_{2} − α′_{1}α′′_{2}}{α′_{1} ^{2} + α′_{2} ^{2}} βx′_{t},

ξ′_{k} = −α_{1}^{−1} α′_{2}βx′_{k}, ξ′_{l} = α_{2}^{−1} α′_{1}βx′_{l}.

Thus we have

λ_{1} = (α′_{1}α′′_{2}− α′′_{1}α′_{2})β^{3}, λ_{2} = α_{1}^{−1} α′_{2}β, λ_{3} = −α_{2}^{−1} α′_{1}β.

(ii) It is easy to see that if at every point of M we have

(5.6)  (p − 1)λ_{2} = −(q − 1)λ_{3}

then the second fundamental tensor H of M satisfies (1.4) on U_{H} ⊂ M. Evidently, (5.6) yields (p − 1)α_{2}α′_{2} = (q − 1)α_{1}α′_{1}, which is equivalent to

α_{2} = \sqrt{c + \frac{q − 1}{p − 1} α_{1}^{2}} ,

where c is a constant. Note that from (4.1) and (5.6) we get easily

tr(H) = λ_{1} + pλ_{2} + qλ_{3}
= λ_{1} + λ_{2} + λ_{3} + (p − 1)λ_{2} + (q − 1)λ_{3} = λ_{1} + λ_{2} + λ_{3} = ϕ.

Thus (1.3) turns into (1.4).

(iii) We consider the case: p = q ⩾ 2. Now (5.6) gives λ_{2} = −λ_{3}. Thus (4.1)–(4.4) and (3.4) yield

ϕ = λ_{1}, ψ = λ_{2}^{2}, ρ = −λ_{1}λ_{2}^{2},

tr(H) = λ_{1}, tr(H^{2}) = λ_{1}^{2} + (n − 1)λ_{2}^{2},

β_{1} = β_{2} = β_{5} = 0, β_{3} = β_{4} = \frac{n − 3}{n − 2} ρ,

γ_{0} = λ_{1}^{2}λ_{2}^{2}, γ_{1} = −λ_{1}λ_{2}^{2}, γ_{2} = −λ_{2}^{2},

ρ_{1} = 0, ρ_{2} = λ_{2}(λ_{1} − λ_{2}), ρ_{3} = −λ_{2}(λ_{1} + λ_{2}),

κ = tr(H))^{2} − tr(H^{2}) = −(n − 1)λ_{2}^{2} = −(n − 1)ψ.

We also have

S^{3} = \frac{2κ}{n − 1} S^{2} − \frac{κ}{n − 1} ( \frac{κ}{n − 1} + (tr(H))^{2}) S,

(tr(H))^{2} = −\frac{(n − 1) tr(S^{3})}{κ^{2}} + \frac{2 tr(S^{2})}{κ} − \frac{κ}{n − 1} .

Conditions (4.7)–(4.9), by making use of the above presented formulas, turn into

R · C = Q(S, R) + \frac{1}{n − 2} Q( S^{2} − \frac{κ}{n − 1} S, G) , (5.7)

C · R = \frac{n − 3}{n − 2} Q(S, R), (5.8)

C · C = \frac{n − 3}{n − 2} ( Q(S, R) + \frac{1}{n − 2} Q( S^{2} − \frac{κ}{n − 1} S, G) ) , (5.9)

respectively. From (5.7) and (5.8) we get immediately

(5.10)  (n − 2)(R · C − C · R) = Q(S, R) + \frac{1}{n − 2} Q( S^{2} − \frac{κ}{n − 1} S, G) .

Thus the difference tensor R · C − C · R is expressed by a linear combination of some Tachibana tensors. We mention that hypersurfaces in spaces of of constant curvature with the tensor R · C − C · R expressed by a linear combination of certain Tachibana tensors were investigated among others in [16, 22, 26, 28, 51]. We also note that (5.9) and (5.10) yield (n − 3)(R · C − C · R) = C · C. Thus the difference tensor R · C − C · R of M is a conformal invariant.

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Department of Applied Mathematics (Received 05 03 2014) Karol Adamiecki University of Economics in Katowice Pułaskiego 25, 40-276 Katowice Poland

    katarzyna.sawicz@ue.katowice.pl