Geometry of Centroaffine Surfaces in $\mathbb{R}^5$

We use Cartan's method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in $\mathbb{R}^5 \setminus \{0\}$ with nondegenerate centroaffine metric. We then give a complete classification of all homogeneous centroaffine surfaces in this class.

In this paper, we consider the case of a 2-dimensional centroaffine surface in R 5 \ {0}. We will use Cartan's method of moving frames to construct a complete set of local invariants for a large class of such surfaces under certain nondegeneracy assumptions (cf. Definition 3.1, Assumption 3.2). In addition, we give a complete classification of the homogeneous centroaffine surfaces in this class-i.e., those that admit a 3-dimensional Lie group of symmetries that acts transitively on an adapted frame bundle canonically associated to the surface (cf. Definition 6.1). Our primary results are Theorem 4.3 and Theorem 5.3, which describe local invariants for centroaffine surfaces with definite and indefinite centroaffine metrics, respectively, and Theorem 6.6, which describes the homogeneous examples.
The paper is organized as follows. In §2, we introduce the basic concepts of centroaffine geometry and centroaffine surfaces in R 5 \ {0}, including the centroaffine frame bundle and the Maurer-Cartan forms. In §3, we begin the method of moving frames and identify firstorder invariants for centroaffine surfaces. Based on these invariants, surfaces may be locally classified as "spacelike," "timelike," or "null." In §4 and §5, we continue the method of moving frames for the spacelike and timelike cases, respectively. (We do not consider the null case here; it may be explored in a future paper.) Finally, in §6 we classify the homogeneous examples in both the spacelike and timelike cases. R 5 \ {0}, g ∈ GL(5, R), we have g · x = gx. The group GL(5, R) may be regarded as a principal bundle over R 5 \ {0}: write an arbitrary element g ∈ GL(5, R) as g = e 0 e 1 e 2 e 3 e 4 , where e 0 , . . . , e 4 ∈ R 5 \ {0} are linearly independent column vectors. Then define the bundle map π : GL(5, R) → R 5 \ {0} by (2.1) π e 0 e 1 e 2 e 3 e 4 = e 0 .
The fiber group H is isomorphic to the stabilizer of the point and this construction endows the manifold R 5 \ {0} with the structure of the homogeneous space GL(5, R)/H. We also think of the bundle π : GL(5, R) → R 5 \ {0} as the centroaffine frame bundle F over R 5 \ {0}. For each point e 0 ∈ R 5 \ {0}, the fiber over e 0 consists of all frames (e 0 , e 1 , e 2 , e 3 , e 4 ) for the tangent space T e 0 (R 5 \ {0})-i.e., all frames for which the first vector in the frame is equal to the position vector.
The Maurer-Cartan forms ω i j on F are defined by the equations (2.2) de i = e j ω j i , 0 ≤ i, j ≤ 4, and they satisfy the Cartan structure equations (For details, see [7] or [1]. ) We are interested in the geometry of 2-dimensional immersionsf : M 2 → R 5 \ {0}; we will use Cartan's method of moving frames to compute local invariants for such immersions under the action of GL(5, R). In order to begin the method of moving frames, consider the induced bundle of centroaffine frames along Σ =f (M ); this is simply the pullback bundle F 0 =f * F over M . A centroaffine frame field along Σ is a section of F 0 -i.e., a smooth map f : M → GL(5, R) such that π • f =f . Throughout the remainder of this paper, we will consider the pullbacks of the Maurer-Cartan forms on F to M via such sections f , and we will suppress the pullback notation.
We will gradually adapt our choice of centroaffine frame fields based on the geometry of Σ. For our first adaptation, consider the subbundle F 1 ⊂ F 0 consisting of all frames for which (e 1 (x), e 2 (x)) span the tangent space Tf (x) Σ for each x ∈ M . A section f : M → F 1 will be called a 1-adapted frame field along Σ. Any two 1-adapted frames (e 0 , . . . , e 4 ), (ẽ 0 , . . . ,ẽ 4 ) based at the same point x ∈ M are related by a transformation of the form (2.4) ẽ 0ẽ1ẽ2ẽ3ẽ3 = e 0 e 1 e 2 e 3 e 4         1 0 0 r 03 r 04 0 a 11 a 12 r 13 r 14 0 a 21 a 22 r 23 r 24 where the 2 × 2 submatrices are elements of GL(2, R). We will denote the group of all matrices of the form in (2.4) by G 1 ; then the bundle F 1 is a principal bundle over M with fiber group G 1 .
If f,f : M → F 1 are two 1-adapted frame fields along Σ, then f,f are related by the equationf (x) = f (x) · g(x) for some smooth function g : M → G 1 , as in equation (2.4). Then the corresponding gl(5, R)-valued Maurer-Cartan forms Ω = [ω i j ],Ω = [ω i j ] on M are related as follows:

Reduction of the structure group and first-order invariants
Now consider the pullbacks of the Maurer-Cartan forms to M via a 1-adapted frame field f . From equation (2.2) for de 0 and the fact that the image of de 0 is spanned by e 1 and e 2 , we have (3.1) ω 0 0 = ω 3 0 = ω 4 0 = 0. Moreover, the 1-forms ω 1 0 , ω 2 0 are semi-basic for the projection π : F 1 → M ; in fact, they form a basis for the semi-basic 1-forms on F 1 .
Differentiating equations (3.1) yields: . Cartan's Lemma (see, e.g., [7] or [1]) then implies that there exist functions h k ij = h k ji , k = 0, 3, 4, on M such that For simplicity of notation, let h k denote the matrix If f,f : M → F 1 are two 1-adapted frame fields related by a transformation of the form (2.4), then we can use equation (2.5) to determine how the corresponding matrices h k ,h k are related. First, it follows from the fact thatẽ 0 = e 0 that dẽ 0 = ẽ 1ẽ2 Then, since we have ẽ 1ẽ2 = e 1 e 2 A, we must have Similar considerations show that Together, equations (3.4), (3.5) imply that Definition 3.1. A centroaffine surface Σ =f (M ) will be called nondegenerate if the matrices h 0 , h 3 , h 4 are linearly independent in Sym 2 (R) at every point of M .
Henceforth, we assume that Σ is nondegenerate; from the group action (3.6) it is clear that this definition is independent of the choice of 1-adapted frame field f : M → F 1 along Σ.
The next step is to use the group action (3.6) to find normal forms for the matrices h 0 , h 3 , h 4 . First consider the action on h 3 , h 4 : it can be written as the composition of two separate actions by the matrices A, B ∈ GL(2, R): If we let P denote the 2-dimensional subspace of Sym 2 (R) spanned by (h 3 , h 4 ), then we see that the action by B preserves P , while A acts on P via It is well-known (see, e.g., [12]) that the action on Sym 2 (R) preserves the indefinite quadratic form up to a scale factor, and that this action has precisely 6 orbits, represented by the matrices 0 0 0 0 , It follows that the induced action (3.7) on the Grassmanian of 2-planes in Sym 2 (R) has precisely 3 orbits, depending on whether the plane P is spacelike, timelike, or null with respect to the quadratic form (3.8). These orbits are represented by the 2-planes (3.9) The type of the plane P spanned by (h 3 , h 4 ) (spacelike, timelike, or null) is preserved by the group action (3.6); thus the type of P at any point x ∈ M is well-defined, independent of the choice of 1-adapted frame field f : M → F 1 . At this point, the method of moving frames dictates that we divide into cases based on the type of P . In order to proceed, we make the following assumption: Assumption 3.2. Assume that Σ has constant type-i.e., that the type of P is the same at every point x ∈ M .
In this paper we will consider only the spacelike and timelike cases; the null case is considerably more complicated and may be explored in a future paper.

The spacelike case
First, suppose that the plane P spanned by (h 3 , h 4 ) is spacelike at every point of M . According to the group action (3.7), we can find a 1-adapted frame field along Σ for which P = P 1 , as in equation (3.9). Furthermore, we can then use the action by B to find a 1-adapted frame field along Σ for which It is straightforward to show that these conditions are preserved by transformations of the form (3.6) with . For simplicity, we will restrict to transformations with A 0 ∈ SO(2, R), λ > 0; this has the advantage of producing a frame bundle whose fiber is a connected Lie group. Thus we will assume that where λ > 0, θ ∈ R. Next, consider the effect of the action (3.6) on h 0 . With A = I 2 and r 03 , r 04 chosen appropriately, we can add any linear combination of h 3 , h 4 to h 0 . Thus we can find a 1adapted frame field for which h 0 is a multiple (nonzero by the nondegeneracy assumption) of the identity matrix I 2 . Then under the action (3.6) with A, B as in (4.1), we havẽ Therefore, we can find a 1-adapted frame field along Σ satisfying the additional condition that h 0 = ±I 2 , and this condition is preserved by transformations of the form (3.6) with A, B as in (4.1), λ = 1, and and r 03 = r 04 = 0.
We will denote the group of all matrices of the form in (4.3) by G 2 ; then the 2-adapted frame fields along Σ are the smooth sections of a principal bundle F 2 ⊂ F 1 over M with fiber group G 2 . The equations (4.2) are equivalent to the condition that the Maurer-Cartan forms associated to a 2-adapted frame field satisfy the conditions (4.4) Differentiating equations (4.4) yields s Lemma to these equations shows that there exists a 1-form α and functions h i jk on F 2 such that If f,f : M → F 2 are two 2-adapted frame fields related by a transformation of the form (4.3), then we can once again use equation (2.5) to determine how the corresponding functions h i jk ,h i jk are related. Some of these relationships are more complicated than others; the most straightforward to compute are those corresponding to the formsω 0 3 ,ω 0 4 . These forms appear as the coefficients ofẽ 0 = e 0 in the equations (2.2) for dẽ 3 , dẽ 4 . By applying equations (4.3) and (4.6), one can show that We will denote the group of all matrices of the form in (4.10) by G 3 ; note that G 3 ∼ = SO(2, R).
Then the 3-adapted frame fields are the smooth sections of a principal bundle F 3 ⊂ F 2 over M with fiber group G 3 . The equations (4.9) are equivalent to the condition that the Maurer-Cartan forms associated to a 3-adapted frame field satisfy the conditions ∧ ω 2 0 = 0, and applying Cartan's Lemma shows that there exist functions h i jk on F 3 such that . Moreover, on F 3 , the last two relations in equations (4.7) simplify to (4.14) At this point, we have canonically associated to any nondegenerate, spacelike centroaffine surface in R 5 \ {0} a frame bundle F 3 over M with fiber group isomorphic to SO(2, R). Thus we have the following theorem: is a nondegenerate, spacelike centroaffine surface. Then the pullbacks of the Maurer-Cartan forms on GL(5, R) to the bundle F 3 of 3-adapted frames on Σ determine a well-defined Riemannian metric I = ω 1 0 2 + ω 2 0 2 on Σ, called the centroaffine metric. Moreoever, there is a well-defined "centroaffine normal bundle" N Σ whose fiber N x Σ at each point x ∈ M is spanned by the vectors (e 3 (x), e 4 (x)) of any 3-adapted frame at x, together with a well-defined Riemannian metric In order to obtain more information about the centroaffine metric, consider the structure equations (2.3) for the semi-basic forms ω 1 0 , ω 2 0 on M . Based on our adaptations, it is straightforward to compute that Therefore, α is the Levi-Civita connection form associated to the centroaffine metric on Σ, and the Gauss curvature K of this metric is determined by the equation The remaining structure equations (2.3) determine relations between the functions h i jk on F 3 and their covariant derivatives with respect to ω 1 0 , ω 2 0 . These relations may be viewed as analogs of the Gauss and Codazzi equations for Riemannian surfaces in Euclidean space. In particular, the analog of the Gauss equation is , while the remainder of the relations are partial differential equations involving the functions h i jk . An analog of Bonnet's Theorem (see [6]) guarantees that, at least locally, any solution of this PDE system gives rise to a nondegenerate, spacelike centroaffine surface, and that this surface is unique up to the action of GL(5, R) on R 5 \ {0}. In particular, the functions h i jk on F 3 form a complete set of local invariants for such surfaces.

The timelike case
Now, suppose that the plane P spanned by (h 3 , h 4 ) is timelike at every point of M . According to the group action (3.7), we can find a 1-adapted frame field along Σ for which P = P 2 , as in equation (3.9). Furthermore, we can then use the action by B to find a 1-adapted frame field along Σ for which Since the latter transformation may be obtained from the former simply by interchanging (e 1 , e 2 ) and (e 3 , e 4 ), we will restrict our attention to transformations of the form (5.1), where A, B are diagonal matrices and B = A 2 .
Next, consider the effect of the action (3.6) on h 0 . With A = I 2 and r 03 , r 04 chosen appropriately, we can add any linear combination of h 3 , h 4 to h 0 . Thus we can find a 1adapted frame field for which h 0 is a multiple (nonzero by the nondegeneracy assumption) of the matrix 0 1 1 0 . Then under the action (3.6) with A, B as in (5.1), we havẽ Thus we can find a 1-adapted frame field along Σ satisfying the additional condition that and this condition is preserved by transformations of the form (3.6) with A, B as in (5.1), such that a 11 a 22 = 1 and r 03 = r 04 = 0. For simplicity, we will assume that a 11 > 0; then we can set Differentiating equations (5.5) yields Applying Cartan's Lemma to these equations shows that there exists a 1-form α and functions h i jk on F 2 such that Thus we can find a 2-adapted frame field along Σ satisfying the conditions that We will denote the group of all matrices of the form in (5.11) by G 3 ; note that G 3 ∼ = SO + (1, 1). Then the 3-adapted frame fields are the smooth sections of a principal bundle F 3 ⊂ F 2 over M with fiber group G 3 . The equations (5.10) are equivalent to the condition that the Maurer-Cartan forms associated to a 3-adapted frame field satisfy the conditions (5.12) ω 0 3 = ω 0 4 = 0. Differentiating equations (5.12) yields  In order to obtain more information about the centroaffine metric, consider the structure equations (2.3) for the semi-basic forms ω 1 0 , ω 2 0 on M . Based on our adaptations, it is straightforward to compute that (5.16) dω 1 0 = −α ∧ ω 1 0 , dω 2 0 = α ∧ ω 2 0 . Therefore, α is the Levi-Civita connection form associated to the centroaffine metric on Σ, and the Gauss curvature K of this metric is determined by the equation As in the spacelike case, the remaining structure equations (2.3) determine relations between the functions h i jk on F 3 and their covariant derivatives with respect to ω 1 0 , ω 2 0 . The analog of the Gauss equation is while the remainder of the relations are partial differential equations involving the functions h i jk . As in the spacelike case, any solution of this PDE system locally gives rise to a nondegenerate, timelike centroaffine surface; this surface is unique up to the action of GL(5, R) on R 5 \ {0}, and the functions h i jk on F 3 form a complete set of local invariants for such surfaces.

Homogeneous examples
The goal of this section is to give a complete classification (up to the GL(5, R)-action on R 5 \ {0}) of the homogeneous examples of spacelike and timelike nondegenerate centroaffine surfaces in R 5 \{0}. First we must define precisely what we mean by the term "homogeneous." Because any such surface Σ =f (M ) has a well-defined Riemannian or Lorentzian metric, any symmetry of Σ must preserve the centroaffine metric on Σ and hence must in fact be an isometry of Σ with its centroaffine metric. Furthermore, if the group of symmetries of Σ acts transitively, then the centroaffine metric must have constant Gauss curvature K. As is well-known, the maximal isometry group of any Riemannian or Lorentzian surface has dimension less than or equal to three, and in the maximal case the isometry group acts transitively on the orthonormal frame bundle. Thus we will use the following definition: Definition 6.1. Let F 3 be the bundle of 3-adapted frames along a nondegenerate, spacelike or timelike centroaffine surface Σ =f (M ) in R 5 \ {0}. A diffeomorphism φ : F 3 → F 3 is called a symmetry of Σ if φ * Ω = Ω; i.e., if φ preserves the Maurer-Cartan forms on F 3 . A nondegenerate spacelike or timelike centroaffine surface Σ =f (M ) will be called homogeneous if the group of symmetries of Σ is a 3-dimensional Lie group that acts transitively on F 3 . Remark 6.2. It might also be of interest to consider the slightly less restrictive assumption that Σ has a 2-dimensional group of symmetries that acts transitively on the base manifold M , but we will not consider this scenario here.
Our procedure for classifying the homogeneous examples is as follows. Observe that if Σ is homogeneous, then all the structure functions h i jk must be constant on the bundle F 3 of 3-adapted frames on Σ. When this condition is imposed, the structure equations (2.3) become algebraic relations among the constants h i jk . Given any solution to these relations, the structure equations (2.3) imply that the corresponding Maurer-Cartan form Ω = [ω i j ] takes values in a 3-dimensional Lie algebra g which is realized explicitly as a Lie subalgebra of gl(5, R). Thus Ω is also the Maurer-Cartan form of the connected Lie group G ⊂ GL(5, R) generated by exponentiating g, and this equivalence of the Maurer-Cartan forms on F 3 with those on G implies that F 3 is a homogeneous space for G; indeed, G must be precisely the symmetry group that was assumed to act transitively on F 3 . Now, choose any point f 0 = (e 0 , e 1 , e 2 , e 3 , e 4 ) ∈ F 3 . Recall that we can view f 0 as an element of GL(5, R). The centroaffine surface Σ is equivalent via the GL(5, R)-action to the surface Σ = f −1 0 · Σ, and the bundle F 3 of 3-adapted frames over Σ is given by F 3 = f −1 0 · F 3 . So without loss of generality, we may assume that f 0 is the identity matrix I 5 . With this assumption, the tangent space T f 0 F 3 is equal to the Lie algebra g, and F 3 must in fact be equal to G. Finally, Σ is given by the image of G under the projection (2.1).
In order to carry out this procedure, we consider the spacelike and timelike cases separately.
while the structure equations (2.3) may be written as Substituting (6.1) into equation (6.2) and imposing the condition that all the functions h i jk are constant leads to a system of algebraic equations for the h i jk . A somewhat tedious, but straightforward, computation shows that this system has precisely two solutions, one for = 1 and one for = −1. These are described in the following two examples.
Furthermore, the Gauss equation (4.16) implies that the centroaffine metric has Gauss curvature K = − 1 3 . Denote the matrices in equation (6.3) by M 0 , M 1 , M 2 , respectively, so that These bracket relations imply that the Lie algebra g ⊂ gl(5, R) spanned by (M 0 , M 1 , M 2 ) is isomorphic to so(1, 2). (They also suggest that a more natural basis might be obtained by multiplying each of M 1 , M 2 by √ 3.) Furthermore, it is straightforward to check that g acts irreducibly on R 5 \ {0}. It is well-known (see, e.g., [2]) that so(1, 2) has a unique irreducible 5-dimensional representation and that this representation arises from a (unique) irreducible representation of SO + (1, 2); it follows that the Lie group G ⊂ GL(5, R) corresponding to the Lie algebra g is isomorphic to SO + (1, 2).
The easiest way to compute a local parametrization for G-and hence for Σ-is to compute the 1-parameter subgroups generated by M 0 , M 1 , M 2 and take products of the resulting group elements.
Warning: This must be done carefully in order to ensure that the resulting products cover the entire group G, which in turn guarantees that the resulting parametrization is surjective onto Σ. The subtlety of this issue can already be seen in the standard representation for so(1, 2): the basis has the same bracket relations as (M 0 , √ 3M 1 , √ 3M 2 ), and exponentiating this basis yields the 1-parameter subgroupsḡ Now consider the following two maps f 1 , f 2 : R 3 → SO + (1, 2), which are obtained by multiplying the elementsḡ 0 (t),ḡ 1 (u),ḡ 2 (v) in different orders: It is not difficult to show that f 1 is surjective onto SO + (1, 2), whereas we can see from the middle column that f 2 is not. Thus we must perform this construction with care. Now, the obvious correspondencē defines a Lie algebra isomorphism between the standard representation of so(1, 2) and our Lie algebra g ⊂ gl(5, R). Therefore, the surjectivity of the map f 1 above implies that the analogous map f : R 3 → G will also be surjective onto G, and it follows that the map f = π • f will be surjective onto Σ. The mapf will also turn out to be independent of t, and when regarded as a function of the two variables (u, v) it will define a surjective parametrizationf : R 2 → Σ. With these considerations in mind, define the 1-parameter subgroups Then set It follows from the discussion above thatf is a surjective map onto Σ. Moreover, it is straightforward to check that the tangent vectorsf u ,f v are linearly independent for all (u, v) ∈ R 2 ; thereforef parametrizes a smooth surface Σ ⊂ R 5 \ {0}, as expected. Topologically, Σ is diffeomorphic to a plane; this is to be expected, as F 3 is isomorphic to the orthonormal frame bundle of the hyperbolic plane H 2 , with π as the projection map.
Furthermore, the Gauss equation (4.16) implies that the centroaffine metric has Gauss curvature K = 1 3 . As in the previous example, denote the matrices in equation (6.7) by M 0 , M 1 , M 2 , respectively. Then we have (6.8) [ These bracket relations imply that the Lie algebra g ⊂ gl(5, R) spanned by (M 0 , M 1 , M 2 ) is isomorphic to so(3, R). Furthermore, it is straightforward to check that g acts irreducibly on R 5 \ {0}. Similarly to the previous example, so(3, R) has a unique irreducible 5-dimensional representation, and this representation arises from a (unique) irreducible representation of SO(3, R); it follows that the Lie group G ⊂ GL(5, R) corresponding to the Lie algebra g is isomorphic to SO(3, R). We will compute a local parametrization for Σ as in the previous example: compute the 1parameter subgroups of G generated by M 0 , M 1 , M 2 and take products of the resulting group elements. Ensuring the surjectivity of the resulting parametrization is easier than in the previous example. First, observe that a basis (M 0 ,M 1 ,M 2 ) for the standard representation of so(3, R) with the same bracket relations as (M 0 , Exponentiating this basis yields the 1-parameter subgroups Then the map f : is easily seen to be surjective onto SO(3, R). Thus the analogous map f : R 3 → G will be surjective onto G, and the mapf = π • f will be surjective onto Σ. So, define the 1-parameter subgroups Then set It follows from the discussion above thatf is a surjective map onto Σ, and we see thatf is also independent of t. Unlike in the previous example, the tangent vectorsf u ,f v are not linearly independent for all (u, v) ∈ R 2 ; indeed,f u = 0 whenever v is an odd multiple of π 2 . Nevertheless, the restriction off to some neighborhood of the point (u, v) = (0, 0) is a smooth embedding, and then homogeneity implies that Σ is smooth everywhere. Topologically, Σ is diffeomorphic to a sphere; this is to be expected, as F 3 is isomorphic to the orthonormal frame bundle of the unit sphere S 2 , with π as the projection map.
(The explicit expressions for these group elements are each too large to fit on one line and are not particularly enlightening.) Then set (6.16)f (u, v, t) = π (g 1 (u) · g 2 (v) · g 0 (t)) It follows from the discussion above thatf is a surjective map onto Σ, and we see thatf is also independent of t. Moreover, it is straightforward to check that the tangent vectors f u ,f v are linearly independent for all (u, v) ∈ R 2 ; thereforef parametrizes a smooth surface Σ ⊂ R 5 \ {0}, as expected.
In this case, F 3 is isomorphic to the orthonormal frame bundle of the timelike surface S 2,1 consisting of all spacelike unit vectors in R 2,1 . This surface is a hyperboloid of one sheet, and so we might expect that Σ would be diffeomorphic to a cylinder. However, if we regard the domain of the parametrization (6.16) as S 1 × R, we see that the map (6.16) is invariant under the transformation (6.17) (u, v) → (u + π, −v).
Projections of Σ to the (x 1 , x 2 , x 0 ), (x 1 , x 2 , x 3 ), (x 1 , x 2 , x 4 ), (x 1 , x 3 , x 0 ), and (x 1 , x 4 , x 0 ) coordinate 3-planes are shown in Figure 3. We collect the results of this section in the following theorem: Theorem 6.6. Letf : M → R 5 \{0} be a centroaffine immersion whose image Σ =f (M ) is a homogeneous, nondegenerate, spacelike or timelike centroaffine surface. Then Σ is equivalent via the GL(5, R)-action on R 5 \ {0} to one of the following: • the immersion of the hyperbolic plane H 2 of Example 6.3; • the immersion of the unit sphere S 2 of Example 6.4; • the immersion of the Lorentzian surface S 2,1 of Example 6.5.