Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 1, 109-117.

Centroaffinely Homogeneous Surfaces in R^{3}

Huili Liu* and Changping Wang**

Department of Mathematics, Northeast University Liaining, PRC-110006 Shenyang Nankai Institut of Mathematics, Nankai University PRC-300071 Tianjin

Abstract. A nondegenerate centroaffine surface in R^{3} is called homogeneous, if for any two points p and q on the surface there exists a centroaffine transformation in R^{3} which takes the surface to itself and takes p to q. In this paper all centroaffinely homogeneous surfaces in R^{3} are determined up to centroaffine transformations. MSC 1991: 53 A 15

1. Introduction

In centroaffine differential geometry we study the properties of hypersurfaces in R^{n+1} that are invariant under the centroaffine group Γ in R^{n+1}. Γ is the subgroup of the affine transformation group in R^{n+1} which keeps the origin 0 ∈ R^{n+1} invariant. It is wellknown that for any hypersurface immersion x : M → R^{n+1} such that x is transversal to x_{∗}(T M) there exists a Γ-invariant symmetric bilinear form G on T M. x is then called a nondegenerate centroaffine hypersurface if G is nondegenerate.

An interesting class of centroaffine hypersurfaces in R^{n+1} is that of the so-called centroaffinely homogeneous hypersurfaces. A hypersurface in R^{n+1} is called centroaffinely homogeneous, if for any two points p and q on the hypersurface there exists a centroaffine transformation in R^{n+1} which takes the hypersurface to itself and takes p to q. Any homogeneous proper equiaffine sphere is evidently centroaffinely homogeneous if we move its center to the origin. So we have plenty examples of centroaffinely homogeneous hypersurfaces in R^{n+1}, which can be found in H. Guggenheimer [2], K. Nomizu and T. Sasaki [4],

* Supported by the DFG-project ”Affine Differential Geometry” at the TU Berlin*** ** Partially supported by the DFG-project ”Affine Differential Geometry at the TU Berlin*** *** Both authors’ address at time: Technische Universität Berlin, FB 3 - Mathematik, MA 8-3, Straße des 17. Juni 136, D-10623 Berlin

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1994 Heldermann Verlag, Berlin

110 H.Liu and C.Wang: Centroaffinely Homogeneous Surfaces in R^{3}

F.Dillen and L. Vrancken [1]. Some examples of centroaffinely homogeneous hypersurfaces which are not equiaffine spheres were given by A. M. Li and the second author of this paper in [3].

Our purpose in this paper is to classify the centroaffinely homogeneous surfaces in R^{3}. We obtain the following classification theorem:

Theorem: Let x : M → R^{3} be a nondegenerate centroaffinely homogeneous surface. Then x is centroaffinely equivalent to one of the following surfaces in R^{3}:

x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1; (i)

x_{1}^{α}x_{2}^{β}x_{3}^{γ} = 1, α ≠ 0, β > 0, γ > 0, α + β + γ ≠ 0; (ii)

(iii) exp(−α arctan \frac{x_{1}}{x_{2}} )(x_{1}^{2} + x_{2}^{2})^{β}x_{3}^{γ} = 1, α^{2} + β^{2} ≠ 0, γ > 0, 2β + γ ≠ 0;

(iv) x_{3} = −x_{1}(α log x_{1} + β log x_{2}), β > 0, α + β ≠ 0; x_{2}x_{3} = ±x_{1}^{2} + x_{2}^{α}, α ≠ 1, 2; (v)

where α, β, γ are constants.

2. The centroaffine surfaces in R^{3}

Let x : M → R^{3} be an immersion of surface and [ , , ] the standard determinant in R^{3}. x is said to be a centroaffine surface if x is always transversal to the tangent plane x_{∗}(T M) at each point of M in R^{3}. We define a symmetric bilinear form G on T M by

(2.1)  G = − ∑_{i,j=1}^{2} \frac{[e_{1}(x), e_{2}(x), e_{i}e_{j}(x)]}{[e_{1}(x), e_{2}(x), x]} θ^{i} ⊗ θ^{j},

where {e_{1}, e_{2}} is a local basis for T M with the dual basis {θ^{1}, θ^{2}}. Note that G is globally defined. A centroaffine surface x is said to be nondegenerate if G is nondegenerate. We call G the centroaffine metric of x. From the equations

(2.2)  e_{i}e_{j}(x) = ∑_{k} Γ_{ij}^{k}e_{k}(x) − G_{ij}x, i, j = 1, 2,

we get the so-called induced connection ∇ = {Γ_{ij}^{k}}. It is clear that the centroaffine metric and the induced connection ∇ are invariant under the centroaffine transformation.

Let ˆ∇ be the Levi-Civita connection of G. We define the 3-form C by

(2.3)  Γ_{ij}^{k} − Γ̂_{ij}^{k} = C_{ij}^{k}, C_{ijk} = ∑_{m} G_{km}C_{ij}^{m}, i, j, j, m = 1, 2.

We know that C = ∑ C_{ijk}θ^{i}θ^{j}θ^{k} is the Fubini-Pick form for x which is totally symmetric. The Chebyshev vector field and the Chebyshev form are defined by

(2.4)  T = ∑_{j} T^{j}e_{j} = \frac{1}{2} ∑_{i} G^{ik}C_{ik}^{j}e_{j},

(2.5)  T^{♯} = ∑_{ij} G_{ij}T^{i}θ^{j}.

By (2.2) and (2.3) we have 2T^{♯} = d(log |[e_{1}, e_{2}, x]|) for an orthonormal basis {e_{1}, e_{2}} ([8]). As is well-known T and T^{♯} are centroaffine invariants.

Let x : M → R^{3} be a centroaffinely homogeneous surface. Since G and T are centroaffine invariants, by homogeneity, we know that G(T, T) = constant. Thus we have the following three cases: (i) T ≡ 0; (ii) G(T, T) = 0 but T ≠ 0; (iii) G(T, T) ≠ 0.

If T ≡ 0, x is a proper homogeneous equiaffine sphere in R^{3} centered at 0 ∈ R^{3}. From [2] and [4] we know that x is one of the following surfaces:

\begin{aligned}x_{1}^{2} + x_{2}^{2} + x_{3}^{2} & = 1; \\ x_{1}^{2} + x_{2}^{2} − x_{3}^{2} & = ±1; \\ x_{1}x_{2}x_{3} & = 1 \\ (x_{1}^{2} + x_{2}^{2})x_{3} & = 1. \end{aligned}

From [3] we have the following examples of centroaffinely homogeneous surfaces not being equiaffine spheres:

Example 2.1: The surface defined by

(2.6)  x_{1}^{α}x_{2}^{β}x_{3}^{γ} = 1, α ≠ 0, β > 0, γ > 0, α + β + γ ≠ 0,

is a homogeneous centroaffine surface in R^{3}. Its group is

\{( \begin{matrix}e^{γu} & 0 & 0 \\ 0 & e^{γv} & 0 \\ 0 & 0 & e^{−αu−βv} \end{matrix} )\}

which acts on the point ^{t}(1, 1, 1), and

G = −\frac{(α^{2} + αγ)}{(α + β + γ)γ} du^{2} − 2\frac{αβ}{(α + β + γ)γ} dudv − \frac{(β^{2} + βγ)}{(α + β + γ)γ} dv^{2};

∥T ∥^{2} = \frac{1}{4}(6 − \frac{α}{β} − \frac{β}{α} − \frac{β}{γ} − \frac{γ}{β} − \frac{γ}{α} − \frac{α}{γ}).

Example 2.2: The surface defined by

(2.7)  exp(−α arctan \frac{x_{1}}{x_{2}} )(x_{1}^{2} + x_{2}^{2})^{β}x_{3}^{γ} = 1, α^{2} + β^{2} ≠ 0, γ > 0, 2β + γ ≠ 0,

is a homogeneous centroaffine surface in R^{3}. Its group is

\{( \begin{matrix}e^{γu} ( \begin{matrix}cos γv & sin γv \\ − sin γv & cos γv \end{matrix} ) & 0 \\ 0 & e^{−2βu+αv} \end{matrix} )\} .

which acts on the point ^{t}(0, 1, 1), and

G = \frac{2β}{γ} du^{2} + 2\frac{α}{γ}dudv + \frac{2βγ − α^{2}}{(2β + γ)γ} dv^{2};

∥T ∥^{2} = γ^{3}(4α^{2}γ + 8β^{2}γ − 4β^{3} − α^{2}β − 4βγ^{2});

Example 2.3: The surface defined by

x_{3} = −x_{1}(α log x_{1} + β log x_{2}), β > 0, α + β ≠ 0,

is a homogenous centroaffine surface in R^{3}. Its group is

\{( \begin{matrix}e^{u} & 0 & 0 \\ 0 & e^{v} & 0 \\ −e^{u}(αu + βv) & 0 & e^{u} \end{matrix} )\} ,

which acts on the point ^{t}(1, 1, 0), and

G = \frac{α}{α + β} du^{2} + 2\frac{β}{α + β} dudv − \frac{β}{α + β} dv^{2};

∥T ∥^{2} = \frac{8β − α}{4β}.

3. The centroaffinely homogeneous surfaces in R^{3} with G(T, T) = 0 and T ≠ 0

Let x : M → R^{3} be a centroaffinely homogeneous surface in R^{3} with G(T, T) = 0 and T ≠ 0. We define e_{1} = T and e_{2} satisfying G(e_{1}, e_{2}) = 1, G(e_{2}, e_{2}) = G(e_{1}, e_{1}) = 0. Then we have the structure equations

(3.1)  dx = ∑_{i} θ_{i}e_{i},

(3.2)  de_{1} = θ_{11}e_{1} + θ_{12}e_{2} − θ_{2}x,
(3.3)  de_{2} = θ_{21}e_{1} + θ_{22}e_{2} − θ_{1}x.

H.Liu and C.Wang: Centroaffinely Homogeneous Surfaces in R^{3} 113

From (3.1), (3.2) and (3.3), we get

(3.4)  dθ_{1} = θ_{1} ∧ θ_{11} + θ_{2} ∧ θ_{21},
(3.5)  dθ_{2} = θ_{1} ∧ θ_{12} + θ_{2} ∧ θ_{22}.

Since

d(log |[e_{1}, e_{2}, x]|) = 2T^{♯} = 2T_{1}θ_{1} + 2T_{2}θ_{2} and G(T, e_{i}) = T^{♯}(e_{i}) = T_{i},

we get

T = T_{2}e_{1} + T_{1}e_{2} = e_{1}, then T_{2} = 1 and T_{1} = 0,

so we have

(3.6)  d(log |[e_{1}, e_{2}, x]|) = 2θ_{2}.

From (3.1), (3.2) and (3.3), we get

(3.7)  dθ_{11} − θ_{12} ∧ θ_{21} + θ_{2} ∧ θ_{1} = 0,
(3.8)  dθ_{12} − θ_{11} ∧ θ_{12} − θ_{12} ∧ θ_{22} = 0,
(3.9)  −dθ_{2} + θ_{11} ∧ θ_{2} + θ_{12} ∧ θ_{1} = 0,
(3.10)  dθ_{21} − θ_{21} ∧ θ_{11} − θ_{22} ∧ θ_{21} = 0,
(3.11)  dθ_{22} − θ_{21} ∧ θ_{12} + θ_{1} ∧ θ_{2} = 0,
(3.12)  −dθ_{1} + θ_{21} ∧ θ_{2} + θ_{22} ∧ θ_{1} = 0.

From (3.5), (3.6) and (3.9),we have

(3.13)  θ_{1} ∧ θ_{12} + θ_{2} ∧ θ_{22} = 0, θ_{12} ∧ θ_{1} + θ_{11} ∧ θ_{2} = 0.

Then by the Cartan’s Lemma

(3.14)  θ_{11} = αθ_{1} + νθ_{2}, θ_{12} = βθ_{1} + αθ_{2}, θ_{22} = αθ_{1} + δθ_{2}.

Let dθ_{1} = ρθ_{1} ∧ θ_{2}, then (3.4) and (3.12) yield

\begin{aligned}ρθ_{1} ∧ θ_{2} & = νθ_{1} ∧ θ_{2} + θ_{2} ∧ θ_{21}, \\ ρθ_{1} ∧ θ_{2} & = −δθ_{1} ∧ θ_{2} + θ_{21} ∧ θ_{2}, \end{aligned}

hence

ρ = \frac{1}{2}(ν − δ),
2θ_{21} ∧ θ_{2} = (ν + δ)θ_{1} ∧ θ_{2},
(3.15)  θ_{21} = \frac{1}{2}(ν + δ)θ_{1} + φθ_{2}.

Since x is homogeneous and {θ_{1}, θ_{2}} is invariant under all centroaffine transformations which take the surface to itself, we know that all coefficients of (3.14) and (3.15) are constants. Moreover, (3.7), (3.14) and (3.15) yield

(3.16)  αν − βφ = 1.

(3.8) and (3.14) yield

(3.17)  βρ = 0.

(3.10), (3.14) and (3.15) yield

(3.18)  (ν + δ)ρ = 0.

(3.11), (3.14) and (3.15) yield

(3.19)  βφ − αδ + 1 = 0.

If ρ ≠ 0, (3.17) and (3.18) become β = 0 and ν + δ = 0. (3.16) and (3.19) become αν = 1 and αδ = 1. So ν = −δ and ν = δ. But ν ≠ 0, δ ≠ 0. Therefore we obtain ρ = 0, ie. dθ_{1} = dθ_{2} ≡ 0. From (2.3), (3.14) and (3.15) we get C_{ijk} = constant. From [3] we know that x is one of the surfaces given by Example 2.1, 2.2 and 2.3.

4. The centroaffinely homogeneous surfaces in R^{3} with G(T, T) ≠ 0

Let x : M → R^{3} be a centroaffinely homogeneous surface in R^{3} with G(T, T) ≠ 0. Since ∥T ∥^{2} = G(T, T) = constant ≠ 0, we define e_{1} by

(4.1)  e_{1} = ∥T ∥^{−1}T, e_{2} ⊥ e_{1}, G(e_{1}, e_{2}) = ε_{i}δ_{ij},

where ε_{i} = ±1. Let {θ_{1}, θ_{2}} be the dual basis of {e_{1}, e_{2}}. Then we have the structure equations

(4.2)  dx = ∑_{i} θ_{i}e_{i},
(4.3)  de_{i} = ∑_{j} θ_{ij}e_{j} − ε_{i}θ_{i}x.

Differentiating (4.2) and (4.3) yield

(4.4)  dθ_{i} = ∑_{j} θ_{j} ∧ θ_{ji},
(4.5)  dθ_{i} = ε_{i} ∑_{j} ε_{j}θ_{ij} ∧ θ_{j},
(4.6)  dθ_{ij} = ∑_{k} θ_{ik} ∧ θ_{kj} − ε_{i}θ_{i} ∧ θ_{j}.

H.Liu and C.Wang: Centroaffinely Homogeneous Surfaces in R^{3} 115

From 2T^{♯} = d(log |[e_{1}, e_{2}, x]|) = 2 ∑_{i} T_{i}θ_{i} = 2ε_{1}∥T ∥θ_{1}, we have dθ_{1} = 0. Then

(4.7)  ∑_{j} θ_{j1} ∧ θ_{j} = 0, ∑_{j} ε_{j}θ_{1j} ∧ θ_{j} = 0.

By the Cartan’s Lemma

(4.8)  θ_{11} = λθ_{1} + μθ_{2}, θ_{21} = μθ_{1} + ε_{1}ε_{2}βθ_{2}, θ_{12} = ε_{1}ε_{2}μθ_{1} + αθ_{2}.

Set dθ_{2} = ρθ_{1} ∧ θ_{2}. From (4.4) and (4.5) we get

\begin{aligned}ρθ_{1} ∧ θ_{2} & = θ_{1} ∧ θ_{12} + θ_{2} ∧ θ_{22}, \\ ρθ_{1} ∧ θ_{2} & = ε_{1}ε_{2}θ_{21} ∧ θ_{1} + θ_{22} ∧ θ_{2}. \end{aligned}

Thus we obtain

(4.9)  ρ = \frac{1}{2}(α − β),
(4.10)  θ_{22} = \frac{1}{2}(α + β)θ_{1} + δθ_{2}.

By the homogeneity of the surface, we know that all λ, μ, α, β, δ, ρ here are constants.

From dθ_{11} = θ_{12} ∧ θ_{21}, we have ρμ = μβ − μα = −2μρ, i.e.

(4.11)  μρ = 0.

If ρ = 0, then dθ_{1} = dθ_{2} = 0. From (2.3), (4.8) and (4.10) we get C_{ijk} = constant for all i, j, k. By [3] we know that x is one of the surfaces given by Example 2.1, 2.2 and 2.3.

Suppose ρ ≠ 0. From (4.11) μ = 0. So dθ_{22} = θ_{21} ∧ θ_{12} yields δ = 0. Thus we have

(4.12)  θ_{11} = λθ_{1}, θ_{12} = αθ_{2}, θ_{21} = ε_{1}ε_{2}βθ_{2}, θ_{22} = \frac{1}{2}(α + β)θ_{1}.

From dθ_{12} = θ_{11} ∧ θ_{12} + θ_{12} ∧ θ_{22} − ε_{1}θ_{1} ∧ θ_{2} we get

(4.13)  α^{2} − λα + ε_{1} = 0.

From dθ_{21} = θ_{21} ∧ θ_{11} + θ_{22} ∧ θ_{21} + ε_{2}θ_{1} ∧ θ_{2} we get

(4.14)  β^{2} − λβ + ε_{1} = 0.

Since ρ = \frac{1}{2}(α − β) ≠ 0, we obtain

(4.15)  λ^{2} − 4ε_{1} > 0.

Therefore

(4.16)  α = \frac{λ}{2} ± \frac{\sqrt{λ^{2} − 4ε_{1}}}{2} , β = \frac{λ}{2} ∓ \frac{\sqrt{λ^{2} − 4ε_{1}}}{2} ,

ρ^{2} = (\frac{1}{2}(α − β))^{2} = \frac{1}{4}(λ^{2} − ε_{1}) > 0.

Set θ_{1} = du, θ_{2} = e^{ρu}dv, e_{1} = x_{u}, e_{2} = e^{−ρu}x_{v}. Then from (4.3) by a direct calculation we get

(4.17)  x_{uu} = λx_{u} − ε_{1}x.
(4.18)  x_{uv} = αx_{v},
(4.19)  x_{vv} = ε_{1}ε_{2}βe^{2ρu}x_{u} − ε_{2}e^{2ρu}x.

From (4.17) and (4.16) we have

(4.20)  x = f(v)e^{αu} + g(v)e^{βu}.

(4.20) and (4.18) yield

x_{uv} = αf′(v)e^{αu} + βg′(v)e^{βu} = αf′(v)e^{αu} + αg′(v)e^{βu},

but α ≠ β, hence g′(v) ≡ 0 and g(v) ≡ c is a constant vector in R^{3}. Thus we have

(4.21)  x = f(v)e^{αu} + ce^{βu}.

(4.21) and (4.19) yield

x_{vv} = f′′(v)e^{αu} = ε_{1}ε_{2}βe^{2ρu}(αf(v)e^{αu} + βce^{βu}) − ε_{2}e^{2ρu}(f(v)e^{αu} + ce^{βu})
= αβε_{1}ε_{2}f(v)e^{(2ρ+α)u} + β^{2}ε_{1}ε_{2}ce^{αu} − ε_{2}f(v)e^{(2ρ+α)u} − ε_{2}ce^{αu}
= ε_{1}ε_{2}c(β^{2} − ε_{1})e^{αu},

so we get f′′(v) = ε_{1}ε_{2}(β^{2} − ε_{1})c, i.e.

f(v) = \frac{1}{2}ε_{1}ε_{2}(β^{2} − ε_{1})cv^{2} + av + b

for some constant vectors a, b ∈ R^{3}. Therefore we obtain

(4.22)  x = [\frac{1}{2}ε_{1}ε_{2}(β^{2} − ε_{1})cv^{2} + av + b]e^{αu} + ce^{βu}

= (a, b, c) ( \begin{matrix}ve^{αu} \\ e^{αu} \\ \frac{1}{2}ε_{1}ε_{2}(β^{2} − ε_{1})v^{2}e^{αu} + e^{βu} \end{matrix} ) ,

where a, b and c are constant vectors in R^{3}. Thus up to a centroaffine transformation we have

(4.23)  x = (ve^{αu} , e^{αu} , kv^{2}e^{αu} + e^{βu})

for k = \frac{1}{2}ε_{1}ε_{2}(β^{2} − ε_{1}). It is a homogeneous surface with the group

\{e^{αu} ( \begin{matrix}e^{−ρu} & v & 0 \\ 0 & 1 & 0 \\ 2kve^{−ρu} & kv^{2} & e^{−2ρu} \end{matrix} )\}

which acts on the point ^{t}(0, 1, 1). Its centroaffine metric is given by

G = ε_{1}du^{2} + ε_{2}e^{(α−β)u}dv^{2},

which has (nonzero) constant curvature. Its Chebyshev vector field satisfies

∥T ∥^{2} = ε_{1} \frac{9}{16}(α + β)^{2}.

Its Pick invariant

J = \frac{1}{2} ∑_{ijkl}^{2} G^{ij}C_{ik}^{l}C_{jl}^{k} = \frac{7}{8}ε_{1}(α + β)^{2}.

Using the Cartesian coordinates (x_{1}, x_{2}, x_{3}) we can write the surface (4.23) as

(4.24)  x_{2}x_{3} = kx_{1}^{2} + x_{2}^{ε_{1}β^{2}+1}.

Since k ≠ 0 and β ≠ 0, it is centroaffinely equivalent of one of the surfaces given by (v) in the theorem in [1].

Acknowledgements. We would like to thank Professor U. Simon for helpful discussions.

References

[1] F. Dillen and L. Vrancken: Calabi-type composition of affine spheres. In: Geometry and Topology of Submanifolds V, World Scientific, Singapore etc. (1993).

[2] H. Guggenheimer: Differential Geometry, Mc Graw-Hill, New York (1963).

[3] A. M. Li and C. P. Wang: Canonical centroaffine hypersurfaces in R^{3}, Results in Mathematics 20 (1991), 660-681.

[4] K. Nomizu and T. Sasaki: A new model of unimodular-affinely homogeneous surfaces, Manuscripta Math. 73 (1991), 39-44.

[5] K. Nomizu and T. Sasaki: On the classification of projectivly homogeneous surfaces, Results in Mathematics 20 (1991), 698-724.

[6] R. Walter: Centroaffine differential geometry: submanifolds of codimension 2, Results in Mathematics 13 (1988), 386-402.

[7] R. Walter: Compact centroaffine spheres of codimension 2, Results in Mathematics 20 (1991), 775-788.

[8] C. P. Wang: Centroaffine minimal Hypersurfaces in R^{n+1}, Geom. Dedicata. (to appear).