Lobachevskii Journal of Mathematics Vol. 3, 1999, 19–38 \begin{matrix}c \\ ◯ \end{matrix}B. Doubrov and B. Komrakov

B. Doubrov and B. Komrakov

CLASSIFICATION OF HOMOGENEOUS SUBMANIFOLDS IN HOMOGENEOUS SPACES

(submitted by B.N.Shapukov)

Abstract. We develop algebraic methods for study of homogeneous submanifolds in homogeneous spaces. This includes the description of stationary subgroups of k-jets of homogeneous submanifolds, algebraic version of Cartan’s method of the moving frame, description of subalgebras corresponding to symmetry groups of homogeneous submanifolds. We also develop several algebraic techniques for classification of homogeneous submanifolds. As application we present the results of such classifications in low-dimensional affine and projective geometries.

1. Introduction

This paper is concerned with homogeneous submanifolds in homogeneous spaces. In general, a submanifold L of a homogeneous space M with transitive transformation group G is called (globally) homogeneous if it is an orbit of a certain virtual Lie subgroup H ⊂ G. The totality of homogeneous submanifolds is one of the key attributes of a homogeneous space and plays a central part in the study of arbitrary submanifolds of that homogeneous space. For example, in projective geometry, the tangent space to a submanifold is a hyperplane, i.e., the simplest homogeneous submanifold in projective geometry that best approximates the given submanifold. Similarly, the curvature of plane curves results from their approximation by circles, which are homogeneous curves in Euclidean geometry, while the torsion of space curves is a result of their approximation by helices, which are homogeneous curves in 3-dimensional Euclidean geometry. This follows from the general fact that homogeneous submanifolds are precisely those submanifolds all of whose differential invariants are constant. This leads to the problem of classifying all homogeneous submanifolds in arbitrary homogeneous spaces.

1991 Mathematical Subject Classification. 53A15, 53A20, 53B25.

Key words and phrases. Jet space, symmetry algebra, moving frame, affine geometry, projective geometry.

In low-dimensional Euclidean geometries, homogeneous submanifolds are well known. In the plane, they are precisely the straight lines and circles. The homogeneous curves in 3-space are the straight lines, circles, and helices, whereas the homogeneous surfaces are the planes, cylinders, and spheres. As regards other low-dimensional homogeneous spaces, homogeneous submanifolds have been studied most thoroughly in affine and projective geometries of dimension ≤ 3. As far back as in the last century, Sophus Lie described homogeneous surfaces in 3-dimensional complex projective geometry [9]. A similar classification for real projective geometry is given in [2], [14] (only surfaces with non-vanishing Pick invariant) and [3] (surfaces with vanishing Pick invariant and degenerate surfaces). Homogeneous surfaces in affine 3-space are described in [1], [4] and [5]. For the corresponding classification in unimodular and centroaffine 3-dimensional geometries, see [6, 7, 8, 10, 12, 13]. The techniques in [3, 14] are purely differential geometric: special adapted frames are constructed and the classification is done by integrating the fundamental equations, using the fact that all geometric invariants are constant. Here we give a different proof of this result, based on the language of Lie algebras.

In Section 2, we study jets of homogeneous submanifolds in homogeneous spaces. In particular, we explicitly describe the stabilizers G_{k} of k-jets of homogeneous submanifolds, which form a decreasing chain of subgroups in the Lie group G of motions of the homogeneous space. We prove that this chain always stabilizes.

Along with a global definition of homogeneous submanifolds, there are a number of local versions of this definition. In Section 3, we show that all of these definitions are equivalent and give a straightforward proof that a locally homogeneous submanifold is always an open part of a globally homogeneous submanifold.

In particular, we show that the germs of locally homogeneous submanifolds at a fixed point are uniquely determined by their symmetry algebras. This allows to reduce the problem of classifying homogeneous submanifolds to the purely algebraic problem of classifying subalgebras in Lie algebras.

In Section 4, we discuss Cartan’s method of the moving frame for locally homogeneous submanifolds and show that in this particular case this method reduces, in fact, to a successive construction of a chain of subalgebras in the Lie algebra of the Lie transformation group.

Based on this method, we develop algebraic methods for classifying symmetry algebras of homogeneous submanifolds, using the structure theory of Lie algebras. We illustrate the resulting algorithm by applying it to the case of homogeneous curves in affine and projective geometries.

In Section 6, as an application of this method, we present the results of the classification all locally homogeneous surfaces in 3-dimensional affine and projective geometries, as well as of locally homogeneous hypersurfaces in 4dimensional affine and projective geometries whose symmetry algebras have dimension ≥ 4.

2. Jets of submanifolds

Let M be an m-dimensional manifold, and let L_{1}, L_{2} be two submanifolds of dimension l containing a point p ∈ M.

Definition. The submanifolds L_{1} and L_{2} are said to be k-equivalent (or also to have contact of order k) at p if there exists a local coordinate system (x^{1}, …, x^{l}, y^{1}, …, y^{m−l}) with origin at p such that

1. the submanifolds L_{i} (i = 1, 2) have, in this coordinate system, equations of the form y_{i}^{j} = f_{i}^{j}(x_{1}, …, x_{l}), j = 1, …, m − l, respectively;

2. for each j, the functions f_{1}^{j} and f_{2}^{j} have the same partial derivatives of order ≤ k at the point (0, …, 0).

It is easily verified that the relation of being k-equivalent is well defined and is indeed an equivalence relation. The k-equivalence class of a submanifold L at a point p ∈ L, denoted [L]_{p}^{k}, is called the k-jet of L. The totality J^{k}(M, l) of all k-jets at all points of the manifold M may be endowed with a smooth manifold structure and is called the space of k-jets of l-dimensional submanifolds of the manifold M.

For any k, n such that 1 ≤ k ≤ n, we define a smooth mapping

π_{n,k}: J^{n}(M, l) → J^{k}(M, l), [L]_{p}^{n} ↦ [L]_{p}^{k},

and for any k ≥ 1 we define a mapping

π_{k}: J^{k}(M, l) → M, [L]_{p}^{k} ↦ p.

If L is an arbitrary l-dimensional submanifold in M, then the subset L^{(k)} = {[L]_{p}^{k} | p ∈ L} is a submanifold in J^{k}(M, l), called the k-th prolongation of L. It is clear that the mapping π_{k} maps L^{(k)} diffeomorphically onto L.

The following result, which we shall need later, is given here without proof, as it follows immediately from the definitions.

Lemma 1. If two submanifolds L_{1} and L_{2} are k-equivalent at a point p ∈ M, then the following two conditions are equivalent:

1. L_{1} and L_{2} are (k + 1)-equivalent at the point p;

2. the tangent spaces to the submanifolds L_{1}^{(k)} and L_{2}^{(k)} at the point q = [L_{1}]_{p}^{k} = [L_{2}]_{p}^{k} ∈ J^{k}(M, l) coincide.

Suppose that a certain Lie group G acts smoothly on M. Then this action may be in a natural way extended to the action of G on J^{k}(M, l):

g.[L]_{p}^{k} = [g.L]_{g.p}^{k}.

Let L be an l-dimensional orbit of the virtual Lie subgroup H ⊂ G, and 𝔥 ⊂ 𝔤 the subalgebra corresponding to H. Suppose that a ∈ L. Consider the space J^{k}(M, l) of n-jets of k-dimensional submanifolds of M. Then L determines a point a_{k} = [L]_{a}^{k} of the manifold J^{k}(M, l). Let G_{0} be the stabilizer of a under the action of G on M.

We describe the stabilizer of the point a_{k} under the natural action of the Lie group G on J^{k}(M, l). To do this, we define a chain of subgroups

G = G_{−1} ⊃ G = G_{0} ⊃ ⋯ ⊃ G_{i} ⊃ G_{i+1} ⊃ …

by letting

G_{i+1} = {g ∈ G_{i} | Ad g(𝔥) ⊂ 𝔥 + 𝔤_{i}} , i ≥ 0,

where 𝔤_{i} denotes the subalgebra corresponding to the subgroup G_{i}.

Theorem 1. G_{k} is precisely the stabilizer of the point a_{k} ∈ J^{k}(M, l).

Proof. The proof is by induction on k. We first proof the theorem for k = 1. In this case, the point a_{1} = [L]_{a}^{1} can be identified with the subspace T_{a}L of the tangent space T_{a}M ≡ 𝔤/𝔤_{0}. Since L is an orbit of the subgroup H, the subspace T_{a}L ⊂ 𝔤/𝔤_{0} has the form (𝔤_{0} + 𝔥)/𝔤_{0}. The isotropic action of the group G_{0} on the tangent space T_{a}M can be naturally identified with the adjoint action of G_{0} on 𝔤/𝔤_{0}. Thus the group G_{1} is the subgroup of G_{0} preserving the subspace T_{a}L ⊂ T_{a}M under the isotropic action. With the above identifications, we have

G_{1} = {g ∈ G_{0} | Ad g(𝔥) ⊂ 𝔥 + 𝔤_{0}}

Now suppose that our theorem holds for an arbitrary number k. Consider the submanifold L^{(k)} ⊂ J^{k}(M, l), which is a prolongation of the submanifold L. Then a_{k} ∈ L^{(k)}, and L^{(k)} is again an orbit of the subgroup H under its action on J^{k}(M, l). Then the point a_{k+1} = [L]_{a}^{k+1} can be identified with the subspace T_{a_{k}}L^{(k)} of the space T_{a_{k}}J^{k}(M, l). Therefore, the stabilizer of this subspace under the isotropic action of G_{k} on T_{a_{k}}J^{k}(M, l) is precisely the same as the stabilizer of the point a_{k+1} in J^{k+1}(M, l). But, similar to the case k = 1, the stabilizer of T_{a_{k}}L^{(k)} is equal to

G_{k+1} = {g ∈ G_{k} | Ad g(𝔥) ⊂ 𝔥 + 𝔤_{k} ∀h ∈ 𝔥},

which proves the theorem.

As we shall see in Sections 4 and 5, the above chain of subgroups of G and the corresponding chain of subalgebras play an important part in Cartan’s method of moving frame and in the algebraic methods for classifying homogeneous submanifolds in homogeneous spaces.

3. Locally homogeneous submanifolds

To make our presentation complete, we list in this section several results (proved in [2]) that establish the equivalence of various definitions of locally homogeneous submanifolds and form a foundation of an algebraic approach to the study of homogeneous submanifolds.

3.1. Invariant foliations on homogeneous spaces. Let M = G/G_{0} be a homogeneous space of a Lie group G and o = eG_{0}. Invariant foliations on M are in a one-to-one correspondence with invariant completely integrable distributions on M. Our goal in this subsection is to prove the following auxiliary result.

Lemma 2. Every invariant foliation on M has the form {(gH)G_{0}}_{g∈G}, where H ⊂ G is some virtual (i.e. not necessarily closed) Lie subgroup containing G_{0}. In particular, the fiber of this foliation at a point o is formed by the orbit of o under the action of H^{o} on M.

Proof. Suppose E is the invariant completely integrable distribution corresponding to a certain invariant foliation on M. Let now 𝔤 be the Lie algebra of G, which we shall identify with T_{e}G, and let 𝔤_{0} be a subalgebra of 𝔤 corresponding to the subgroup G_{0}. We consider the natural projection π: G → M, g ↦ gG_{0} and identify the space T_{o}M with 𝔤/𝔤_{0} by means of the surjection d_{e}π: 𝔤 → T_{o}M. Let 𝔥 = d_{e}π^{−1}(E_{o}). Since E is completely integrable, it follows that 𝔥 is a subalgebra of 𝔤. Let H_{0} be the connected virtual Lie subgroup of the Lie group G, corresponding to the subalgebra 𝔥.

From the invariance of the distribution E it follows that the subalgebra 𝔥 is stable under Ad G_{0}. Hence the subgroup H_{0} is stable under the conjugation by the elements of G_{0}. Put H = H_{0}G. Since H_{0} contains the identity component of G, we see that H is a virtual subgroup of G whose identity component coincides with H_{0}. It is now easy to show that π(H) = π(H_{0}) is a maximal integral manifold of the distribution E and has the form HG_{0} = H_{0}G_{0}. Since the foliation is invariant, the rest of its fibers have the form (gH)G_{0} with g ∈ G.

3.2. Definitions of locally homogeneous submanifolds. Let L be an immersed submanifold in M. We assume that the action of G on M is locally effective and identify the Lie algebra 𝔤 of G with a subalgebra of the Lie algebra of all vector fields on M.

Definition. The symmetry algebra of a submanifold L is the subalgebra sym(L) in 𝔤 defined by

sym(L) = {X ∈ 𝔤 | X_{p} ∈ T_{p}L for all p ∈ L}.

For any subalgebra 𝔥 ⊂ 𝔤 and arbitrary point p ∈ M, let 𝔥_{p} denote the subspace T_{p}M of the form {X_{p} | X ∈ 𝔥}. For all points p ∈ L we obviously have sym(L)_{p} ⊂ T_{p}L.

Theorem 2. If L is a connected immersed submanifold in M, then the following four conditions are equivalent:

1. for any point p ∈ L there exists a neighborhood U of p in L and a smooth mapping ϕ: U → G such that ϕ(q).p = q and ϕ(q).U ⊂ L for all q ∈ U;

2. given two arbitrary points p, q ∈ L, there always exist a neighborhood U of p in L and an element g ∈ G such that g.p = q and g.U ⊂ L;

3. the submanifold L is an open subset in some orbit of a connected virtual subgroup H ⊂ G;

4. for any point p ∈ L the following equality holds: sym(L)_{p} = T_{p}L.

Proof. (1) ⇒ (2). We fix an arbitrary point p ∈ L and denote by N the set of all points q in L for which condition (2) is satisfied (i.e., there exists a neighborhood of p in L and an element g ∈ G such that g.p = q and g.U ⊂ L). The set N is nonempty and is open in L. Indeed, it contains the point p itself, and it follows from condition (1) that together with each point q this set contains a certain neighborhood of q. By a similar reasoning, the complement of N in L is also open. Since L is connected, we have N = L and L satisfies condition (2).

(2) ⇒ (3). Consider the natural lift of the action of the Lie group G to J^{k}(M, l). Fix a point p ∈ L and let p_{k} = [L]_{p}^{k}. Let G_{k} denote the stationary subgroup of p_{k}. It is clear that G_{k+1} ⊂ G_{k} for all k ≥ 0. Hence for some sufficiently large n we shall have dim G_{n−1} = dim G_{n}.

Denote by E_{k} the orbit of p_{k} under the action of G on J^{k}(M, l). It immediately follows from (2) that L^{(k)} ⊂ E_{k}. Define

π = π_{n,n−1}|_{E_{n}}: E_{n} → E_{n−1}.

Since the dimensions of the stationary subgroups G_{n} and G_{n−1} are the same, π is a local diffeomorphism.

Consider the subspace E_{p_{k}} = T_{p_{k}}L^{(k)} of the space T_{p_{k}}E_{k} for all k ≥ 0. We shall now show that E_{p_{n}} is invariant under the isotropic action of G_{n} on T_{p_{n}}E_{n}. Indeed, we have

d_{p_{n}}g(E_{p_{n}}) = T_{p_{n}}(g.L)^{(n)} for all g ∈ G_{n}.

Since the submanifolds L and g.L are n-equivalent at p, we have

T_{p_{n−1}}(g.L)^{(n−1)} = E_{p_{k−1}}.

But the mapping d_{p_{n}}π is an isomorphism of the spaces T_{p_{n}}E_{n} and T_{p_{n−1}}E_{n−1} and transforms each of the subspaces E_{p_{n}} and T_{p_{n}}(g.L)^{(n)} into the subspace E_{p_{n−1}}. Hence d_{p_{n}}g(E_{p_{n}}) = E_{p_{n}} for all g ∈ G_{n}. This allows to define a G-invariant distribution E on E_{n} by the formula

E_{q} = d_{p_{n}}g(E_{p_{n}}) for all g ∈ G, g.p_{n} = q.

We show that E is completely integrable. It will actually suffice to show that L^{(n)} is an integral manifold of M. Indeed, let q = π_{n}(q_{n}) for any q_{n} ∈ L^{(n)}. Now let g be an element of G such that g.p = q and g.U ⊂ L for some neighborhood U of p in L. Then g.p_{n} = q_{n} and g.U^{(n)} ⊂ L^{(n)}. It follows that

T_{q_{n}}L^{(n)} = d_{p_{n}}g(T_{p_{n}}L^{(n)}) = d_{p_{n}}g(E_{p_{n}}) = E_{q_{n}}.

Suppose N is the maximal integral manifold of E passing through p. By Lemma 2, there exists a connected Lie subgroup H ⊂ G such that N is the orbit of p_{n} under the action of H on J^{n}(M, l). In addition, L^{(n)} is an open subset in N. Thus L is an open subset of the orbit of p under the action of H on p.

(3) ⇒ (1). Let p be an arbitrary point in the submanifold L, and let N be the orbit of p under the action of H on M. Consider the natural projection π: H → N, g ↦ g.p. Then π is a fibration, and in some neighborhood U of p there exists a section ϕ: U → H of π. We may assume without loss of generality that U ⊂ L. But then the neighborhood U and the mapping ϕ satisfy condition (1).

(3) ⇒ (4). Let 𝔥 be a subalgebra of 𝔤 corresponding to the subgroup H. It is clear that 𝔥 ⊂ sym(L), and 𝔥_{p} = T_{p}L for all p ∈ L. It now easily follows that sym(L)_{p} = T_{p}L for all p ∈ L.

(4) ⇒ (3). Let H be the connected virtual subgroup of G corresponding to the subalgebra sym(L) ⊂ 𝔤, and let p be an arbitrary point of L. It is easy to show that the orbit of p under the action of H on M contains L as an open subset.

Definition. A connected immersed submanifold L in M is called locally homogeneous if it satisfies one of the equivalent conditions of Theorem 2.

4. The method of moving frame for homogeneous submanifolds

Let M be a homogeneous space of a Lie group G, and let G_{0} be the stabilizer of a point o. As before, we identify M with G/G_{0}. Denote by π: G → M the natural projection with π ↦ g.o ≡ gG_{0} for all g ∈ G.

Let 𝔤 be the Lie algebra of G, and 𝔤_{0} the subalgebra of 𝔤 corresponding to G_{0}. We shall identify 𝔤 with T_{e}G and 𝔤/𝔤_{0} with T_{o}M. Suppose that L is a locally homogeneous submanifold in M passing through o.

Consider the problem of constructing a moving frame for L in its modern formulation as presented in [8, 16, 17, 18]. In accordance with [16], we construct a series of 4-tuples (E_{i}, π_{i}, G_{i}, ω_{i}) with i = 0, 1, … that satisfy the following conditions:

a) G_{i} is a subgroup in G_{0};

b) E_{i} is a submanifold in G such that π: E_{i} → L is a principle G_{i}-bundle;

c) ω_{i} is a 𝔤-valued form on E_{i} such that (i) (ω_{i})_{p}: T_{p}E_{i} → 𝔤 is injective for all p ∈ E_{i}; (ii) ω_{i}(X^{∗}) = X for all X ∈ 𝔤_{i}, where 𝔤_{i} is the subalgebra corresponding to G_{i}, and X^{∗} is the fundamental vector field on E_{i} corresponding to X;

(iii) R_{g}^{∗}ω_{i} = Ad g^{−1} ∘ ω_{i} for all g ∈ G_{i}, where R_{g} is the right shift on E_{i}

determined by the element g; (iv) dω_{i} + \frac{1}{2}[ω_{i}, ω_{i}] = 0.

For i = 0, we let E_{0} = π^{−1}(L), π_{0} = π|_{E_{0}}, ω_{0} = ω|_{E_{0}}, where ω is the leftinvariant Maurer-Cartan form on G. For i ≥ 0, the 4-tuple (E_{i+1}, π_{i+1}, G_{i+1}, ω_{i+1}) can be obtained from the 4-tuple (E_{i}, π_{i}, G_{i}, ω_{i}) in such a way that G_{i+1} is a subgroup of G_{i}, E_{i+1} is a reduction of the principal G_{i}-bundle E_{i} to the subgroup G_{i+1} (both the subgroup and reduction will be defined shortly), and ω_{i+1} = ω_{i}|_{E_{i+1}}. It is obvious that the 4-tuple (E_{i+1}, π_{i+1}, G_{i+1}, ω_{i+1}) thus defined satisfies the above conditions.

It now remains to the subgroup G_{i+1} ⊂ G_{i} and the reduction of the G_{i}- bundle E_{i} to G_{i+1}. To do this, we define the notion of structure function f_{i} of a 4-tuple (E_{i}, π_{i}, G_{i}, ω_{i}). Consider the Grassmann manifold Gr_{l}(𝔤/𝔤_{i}) of all l-dimensional subspaces in 𝔤/𝔤_{i}, where l = dim L. The group G_{i} acts naturally on Gr_{l}(𝔤/𝔤_{i}) by means of the adjoined representation induced on 𝔤/𝔤_{i}.

Definition. The function defined by

f_{i}: E_{i} → Gr_{l}(𝔤/𝔤_{i}), p ↦ im(ω_{i})_{p}/𝔤_{i}

is called the structure function of the 4-tuple (E_{i}, π_{i}, G_{i}, ω_{i}).

Lemma 3.

1) The mapping f_{i} is G_{i}-invariant, that is f_{i}(pg) = g^{−1}.f_{i}(p) for all g ∈ G_{i}, p ∈ E_{i};

2) The image of f_{i} lies entirely in a certain orbit of the action of G_{i} on Gr_{l}(𝔤/𝔤_{i}).

Proof. The first assertion follows immediately from the definition of f_{i} and from the property (iii) of ω_{i}. To prove the second assertion of the lemma for i = 0, let l_{g} denote the transformation of M defined by an element g ∈ G, so that l_{g}: p ↦ g.p for all p ∈ M. Using the identification of 𝔤/𝔤_{0} with T_{o}M, prove that

f_{0}(g) = d_{o}l_{g}^{−1}(T_{g.o}L) for all g ∈ G. (1)

Indeed, from the commutative diagram

\begin{matrix}G & ⟶^{L_{g}} & G \\ ^{π}↓ & & ^{π}↓ \\ M & ⟶^{l_{g}} & M \end{matrix}

it follows that the following diagram is commutative:

\begin{matrix}𝔤 ≡ T_{e}G & ⟶^{d_{e}L_{g}} & T_{g}G \\ ^{d_{e}π}↓ & & ^{d_{e}π}↓ \\ 𝔤/𝔤_{0} ≡ T_{o}M & ⟶^{d_{o}l_{g}} & T_{g.o}M \end{matrix}.

Note that E_{0} = π^{−1}(L). The desired formula (1) now follows easily.

Now since the submanifold L is locally homogeneous, we see that for any two points p_{1} = g_{1}.o, p_{2} = g_{2}.o ∈ L with g_{1}, g_{2} ∈ G, there exists an element g ∈ G such that g.p_{1} = p_{2} and d_{p_{1}}l_{g}(T_{p_{1}}L) = T_{p_{2}}L. It follows that

d_{p_{1}}l_{g} ∘ d_{o}l_{g_{1}}(f_{0}(g_{1})) = d_{o}l_{g_{2}}(f_{0}(g_{1})),

so that

f_{0}(g_{2}) = d_{o}l_{g_{2}^{−1}gg_{1}}(f_{0}(g_{1})) = ( Ad g_{2}^{−1}gg_{1})(f_{0}(g_{1})).

For i > 0, consider the manifold M_{i} = G/G_{i} and the image L_{i} ⊂ M_{i} of E_{i} under the natural projection G → M_{i}. Then L_{i} is a locally homogeneous submanifold in M_{i}, and the same reasoning and formulas as in the case i = 0 can be applied to it. This proves the lemma for any i ≥ 0.

Let us fix the point V_{i} = f_{i}(e) ⊂ 𝔤/𝔤_{i} in the image of the structure function f_{i}, let G_{i+1} be equal to the stabilizer of this point under the action of G_{i} on Gr_{l}(𝔤/𝔤_{i}):

G_{i+1} = {g ∈ G_{i} | g.V_{i} = V_{i}}, (2)

and let E_{i+1} = f_{i}^{−1}(V_{i}). It is then clear that E_{i+1} is a reduction of the G_{i}- bundle E_{i} to G_{i+1}, and e ∈ E_{i+1}.

Thus, to construct a moving frame for L, we need to select a family of subspaces V_{i} ⊂ 𝔤/𝔤_{i}. Note that the subalgebra 𝔤_{i+1} corresponding to G_{i+1} can be defined directly from V_{i} using (2), as follows:

𝔤_{i+1} = {X ∈ 𝔤_{i} | [X, Y] + 𝔤_{i} ∈ V_{i} for all Y + 𝔤_{i} ∈ V_{i}}.

Since all the objects in question are finite dimensional, there exists a sufficiently large N ∈ ℕ for which G_{N} = G_{N+1} and E_{N} = E_{N+1}.

Theorem 3.

1) E_{N} is an open subset in some virtual subgroup H of G;

2) L is an open subset in the orbit of H passing through o.

Proof. 1) Since E_{N+1} = E_{N}, the structure function f_{N} is constant on E_{N} and is equal to some subspace V_{N} ⊂ 𝔤/𝔤_{N}. Then, using the equality dω_{N} + \frac{1}{2}[ω_{N}, ω_{N}] = 0, it can be shown that im ω_{N} is a subalgebra. The subalgebra im ω_{N} determines a left-invariant completely integrable distribution on G, and E_{N} is an integral manifold of that distribution. The first assertion of the theorem now follows easily.

The second assertion is immediate from the first.

Notice that the decreasing chain of subgroups G_{i} in G obtained in the process of construction of a moving frame for a homogeneous submanifold is precisely the same as the chain that we constructed in Section 2. This fact can be easily explained. Indeed, it follows from the construction of the moving frame that the 4-tuple (E_{i}, π_{i}, G_{i}, ω_{i}) that we obtain at the i-th stage of the construction depends only on the i-jet of L.

5. Algebraic classification methods

In this section we develop an algorithm for classifying symmetry algebras of homogeneous submanifolds. It will become apparent that this algorithm is none other than the algebraic analog of the method of moving frame discussed in the previous section.

Consider a smooth manifold M with a transitive action of a Lie group G. Let 𝔥 be an arbitrary subalgebra of 𝔤, and H the corresponding connected virtual subgroup of G. Then the orbits of H can be viewed as embedded submanifolds of M. We say that L is an orbit of 𝔥 through a point p ∈ M if L is a connected open submanifold (in the internal topology) of the orbit of H through the point p, and p ∈ L. Of course, an orbit of 𝔥 through p ∈ M is not unique, but any two orbits coincide in a certain neighborhood of p.

Let L be an orbit of 𝔥. Then, obviously, 𝔥 ⊂ sym(L). It is easy to see that L is locally homogeneous. But, generally speaking, sym(L) is bigger than 𝔥. The next theorem determines the whole symmetry algebra of L.

Theorem 4 ([4]). Let o be a fixed point on M, and let L be an orbit of 𝔥 through o. Define a subalgebra 𝔤_{0} ⊂ 𝔤 as follows: 𝔤_{0} = {X ∈ 𝔤 | X_{o} = 0}.

1. The subalgebra sym(L) is the largest subalgebra 𝔞 such that 𝔥 ⊂ 𝔞 ⊂ 𝔤_{0}+𝔥.

2. Consider the series of subalgebras 𝔤_{0} ⊃ 𝔤_{1} ⊃ … such that 𝔤_{n+1} = {x ∈ 𝔤_{n} | [x, 𝔥] ⊃ 𝔤_{n} + 𝔥} for all n ≥ 0, and let 𝔤_{∞} = ∩_{n=0}^{∞}𝔤_{n}. Then

sym(L) = 𝔤_{∞} + 𝔥.

Corollary. The subalgebra 𝔥 ⊂ 𝔤 is a symmetry algebra of some homogeneous submanifold L containing o if and only if 𝔥 satisfies one of the following equivalent conditions:

(1) every subalgebra of 𝔤 that contains 𝔥 and lies in 𝔤_{0} + 𝔥 coincides with 𝔥;

(2) 𝔤_{∞} ⊂ 𝔥.

Definition. We call a subalgebra 𝔥 ⊂ 𝔤 admissible (with respect to the pair (𝔤, 𝔤_{0})) if it satisfies the condition of the corollary.

Let us now develop a method for the classification of all admissible subalgebras 𝔥 ⊂ 𝔤. Suppose that 𝔥 is such a subalgebra. Define the following vector spaces and linear mappings between them:

– let V_{n} = (𝔥+𝔤_{n})/𝔤_{n} ⊂ 𝔤/𝔤_{n} for all n ≥ 0, and V_{∞} = (𝔥+𝔤_{∞})/𝔤_{∞} ⊂ 𝔤/𝔤_{∞};

– let π_{n}: 𝔤 → 𝔤/𝔤_{n} and τ_{n}: 𝔤/𝔤_{n+1} → 𝔤/𝔤_{n} be the natural projections for all n ≥ 0, and π_{∞} is defined in a similar way.

We have the following commutative diagram (here vertical arrows correspond to the natural embeddings):

𝔥 ⟶ V_{∞} ⟶ ⋯ ⟶ V_{1} ⟶ V_{0}
↓  ↓  ↓  ↓  ↓
𝔤 ⟶^{π_{∞}} 𝔤/𝔤_{∞} ⟶ ⋯ ⟶^{τ_{1}} 𝔤/𝔤_{1} ⟶^{τ_{0}} 𝔤/𝔤_{0}

This diagram is finite, i.e. 𝔤_{n} = 𝔤_{n+1} = ⋯ = 𝔤_{∞} and V_{n} = V_{n+1} = … for sufficiently large n.

It is easy to see that these objects satisfy the following conditions:

1. 𝔤_{∞} = 𝔥 ∩ 𝔤_{n} for all n ≥ 0.

2. dim V_{n}=dim V_{∞} for all n ≥ 0.

3. τ_{n}|_{V_{n+1}} is an isomorphism of V_{n+1} to V_{n} for all n ≥ 0.

4. 𝔤_{n+1} = {x ∈ 𝔤_{n} | x.V_{n} ⊂ V_{n}}. (Here we consider 𝔤/𝔤_{n} as a natural 𝔤_{n}-module.)

5. 𝔥 = π_{∞}^{−1}(V_{∞}).

6. The subspace V_{n} for all n ≥ 1 satisfies the following condition:

CS: for every two elements x+𝔤_{n}, y +𝔤_{n} ∈ V_{n}, the element [x, y]+𝔤_{n−1} lies in V_{n−1}.

We say that two subalgebras of 𝔤 are equivalent if they can be transformed into each other by means of the elements of Ad G. For a given subspace V_{n} ⊂ 𝔤/𝔤_{n}, denote by G_{n+1} the subgroup in Ad G consisting of all automorphisms that preserve the subalgebra 𝔤_{n} and induce on 𝔤/𝔤_{n} a transformation preserving V_{n}. From the definition it follows that G_{n+1} also preserves 𝔤_{n+1} and, therefore, induces a group of linear transformations on 𝔤/𝔤_{n+1}.

From the above considerations we obtain the following algorithm for the description (up to equivalence) of all admissible subalgebras in 𝔤 corresponding to locally homogeneous submanifolds of a given dimension m.

1. Describe all subspaces V_{0} of dimension m in 𝔤/𝔤_{0} (up to the transformations induced by G_{0} = {f ∈ Ad G | f(𝔤_{0}) = 𝔤_{0}}).

2. For each subspace V_{n} found above, determine the subalgebra 𝔤_{n+1}, the group G_{n+1}, and the subspace W = τ_{n}^{−1}(V_{n}) in 𝔤/𝔤_{n+1}. If 𝔤_{n+1} ≠ 𝔤_{n}, then describe (up to G_{n+1}) all subspaces V_{n+1} in W such that

– V_{n+1} is a complement of ker τ_{n} = 𝔤_{n}/𝔤_{n+1};

– V_{n+1} satisfies the condition CS.

Then repeat this step again.

3. If 𝔤_{n} = 𝔤_{n+1}, then find the subspace 𝔥 = π_{n}^{−1}(V_{n}) in 𝔤. If 𝔥 is a subalgebra, then Theorem 4 implies that it is a symmetry algebra of a certain homogeneous submanifold. Moreover, all admissible algebras can be obtained in this way.

This algorithm is effective when we want to classify all homogeneous submanifolds L for which dim sym(L) > dim(L), i.e. when 𝔤_{∞} ≠ {0}. If, however, we try to implement the algorithm in the case dim(L) = sym(L), some serious difficulties arise (see examples below). Hence the following alternative algorithm can be used to describe homogeneous submanifolds L for which dim(L) = sym(L):

1. classify all subalgebras 𝔥 of dimension dim L in 𝔤;

2. for each of these subalgebras 𝔥, find the corresponding virtual subgroup H ⊂ G and describe (up to Z(H)) all its orbits of dimension dim L on M;

3. among the submanifolds obtained, select those whose symmetry algebras

coincide with 𝔥. Note that for arbitrary homogeneous spaces, this method is no longer effective. Indeed, it involves classifying subalgebras of arbitrary dimension in a given Lie algebras, which, in itself, is quite a laborious problem, and moreover, a vast majority of subalgebras of dimension higher than dim L are not symmetry algebras for homogeneous spaces.

To illustrated the above algorithms, we shall now provide several simple examples of classifying homogeneous submanifolds.

Example 1. We describe all homogeneous curves in 2-dimensional affine geometry. To do this, we identify the group of affine transformations of the plane with the subgroup

G = \{(\begin{matrix}a_{11} & a_{12} & b_{1} \\ a_{21} & a_{22} & b_{2} \\ 0 & 0 & 1 \end{matrix}) | a_{11}a_{22} − a_{21}a_{12} ≠ 0\}

in GL(3, ℝ). We choose o to be the origin of coordinates on the plane. Then

G_{0} = \{(\begin{matrix}a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{matrix}) | a_{11}a_{22} − a_{21}a_{12} ≠ 0\} .

The corresponding subalgebras 𝔤 and 𝔤_{0} in 𝔤𝔩(3, ℝ) have the form

𝔤 = \{(\begin{matrix}a_{11} & a_{12} & b_{1} \\ a_{21} & a_{22} & b_{2} \\ 0 & 0 & 0 \end{matrix})\} ,

𝔤_{0} = \{(\begin{matrix}a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 0 \end{matrix})\} .

As a first step towards the classification of symmetry algebras of homogeneous curves, we need to describe all 1-dimensional homogeneous subspaces V_{0} in 𝔤/𝔤_{0} up to the adjoint action of G_{0}. In this case it is obvious that G_{0} acts on 𝔤/𝔤_{0} by means of arbitrary linear transformations, and hence we can assume that

V_{0} = \{(\begin{matrix}0 & 0 & x \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{0}\} .

Then the subgroup G_{1} = {g ∈ G_{0} | g.V_{0} = V_{0}} has the form

\{(\begin{matrix}a_{11} & a_{12} & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & 1 \end{matrix}) | a_{11}a_{22} ≠ 0\} .

An arbitrary homogeneous subspace V_{1} in 𝔤/𝔤_{1}, which is mapped onto V_{0} under the natural projection 𝔤/𝔤_{1} → 𝔤/𝔤_{0}, has the form

V_{1}(α) = \{(\begin{matrix}0 & 0 & x \\ αx & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{1}\} , α ∈ ℝ.

The action of the group G_{1} on 𝔤/𝔤_{1} induces the following transformations on the set of subspaces V_{1}(α):

V_{1}(α) ↦ V_{1}(a_{22}α/a_{11}^{2}).

It follows that we can assume that α = 0 or α = 1. We shall consider these two cases separately.

1^{∘}. α = 0: In this case G_{2} = G_{1}, so that the algorithm terminates, and we obtain the following symmetry algebra:

𝔥 = \{(\begin{matrix}a_{11} & a_{12} & x \\ 0 & a_{22} & 0 \\ 0 & 0 & 0 \end{matrix})\} .

The corresponding subgroup H in G has the form

H = \{(\begin{matrix}a & b & c \\ 0 & d & 0 \\ 0 & 0 & 1 \end{matrix}) | a, d ∈ ℝ_{+}^{∗}, b, c ∈ ℝ\} ,

and its orbit H through the point o is L = {(c, 0) | c ∈ ℝ}, i.e., the straight line given by the equation y = 0.

2^{∘}. α = 1: In this case G_{2} has the form

G_{2} = \{(\begin{matrix}a_{11} & a_{12} & 0 \\ 0 & a_{11}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}) | a_{11} ≠ 0\} ,

and we must continue with the algorithm. The desired subspaces V_{2} ⊂ 𝔤/𝔤_{2} have the form

V_{2}(α) = \{(\begin{matrix}αx & 0 & x \\ x & −αx & 0 \\ 0 & 0 & 0 \end{matrix})\} , α ∈ ℝ.

The action of the group G_{2} on 𝔤/𝔤_{2} induces the following transformations on the set of subspaces V_{2}(α):

V_{2}(α) ↦ V_{2}((a_{11}α + a_{12})/a_{11}^{2}).

It follows that we can let α = 0. Then

G_{3} = \{(\begin{matrix}a_{11} & 0 & 0 \\ 0 & a_{11}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}) .\} .

The subspaces V_{3} ⊂ 𝔤/𝔤_{3} have the form

V_{3}(α) = \{(\begin{matrix}0 & αx & x \\ x & 0 & 0 \\ 0 & 0 & 0 \end{matrix})\} , α ∈ ℝ.

The action of G_{3} on 𝔤/𝔤_{3} induces the following transformations on the family of subspaces V_{3}(α):

V_{3}(α) ↦ V_{3}(α/a_{11}^{2}).

Thus we can assume that assume that α = 0, ±1. Consider these three cases one by one.

2.1^{∘}. α = 0: In this case G_{4} = G_{3}, so that the algorithm terminates and we obtain the symmetry algebra

𝔥 = \{(\begin{matrix}a & 0 & x \\ x & 2a & 0 \\ 0 & 0 & 0 \end{matrix}) | a, x ∈ ℝ\} .

The corresponding subgroup H has the form:

H = \{(\begin{matrix}a & 0 & b \\ ab & a^{2} & b^{2}/2 \\ 0 & 0 & 1 \end{matrix}) | a ∈ ℝ_{+}^{∗}, b ∈ ℝ\} .

The orbit of H through o has the form L = {(b, b^{2}/2) | b ∈ ℝ and is the parabola given by y = x^{2}/2.

2.2^{∘}. α = ±1: In this case the group G_{4} is discrete and consists of two elements:

G_{4} = \{E_{3}, (\begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})\} .

Hence the following step will be the last one. Indeed, any 1-dimensional subalgebra 𝔥 which is a symmetry algebra of some homogeneous submanifold must have the form

𝔥_{±}(α) = \{(\begin{matrix}αx & ±x & x \\ x & 2αx & 0 \\ 0 & 0 & 0 \end{matrix})\} , α ∈ ℝ.

To find explicit equations for the corresponding homogeneous curves, we need to construct the subgroups H ⊂ G corresponding to these subalgebras, which, in effect, is equivalent to finding the exponents of the matrices contained in 𝔥_{±}(α). However, since these matrices involve a parameter α, this problem is fairly difficult.

Because of this, in order to find explicit equations for homogeneous curves with 1-dimensional symmetry algebras, we shall use an alternative method based on the description of all 1-dimensional subalgebras in 𝔰𝔩(3, ℝ) up to conjugation. Essentially, the latter problem is equivalent to finding all Jordan forms of 3×3 matrices. Without going into detail, we present here the resulting list of subalgebras:

a) \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})\};

b) \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix})\};

c) \{(\begin{matrix}x & x & 0 \\ 0 & x & 0 \\ 0 & 0 & 0 \end{matrix})\};

d) \{(\begin{matrix}x & 0 & 0 \\ 0 & αx & 0 \\ 0 & 0 & 0 \end{matrix})\}, α ∼ 1/α;

e) \{(\begin{matrix}x & 0 & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix})\};

f) \{(\begin{matrix}αx & x & 0 \\ −x & αx & 0 \\ 0 & 0 & 0 \end{matrix})\}, α ∼ −α.

The equivalence relation α ∼ 1/α in the case d means that the two subalgebras corresponding to the inverse values α and 1/α are conjugate to one another. Similarly in the case f, the two subalgebras corresponding to the values α and −α are conjugate.

Note that in contrast to the case of the subalgebras 𝔥_{±}(α) discussed earlier, the subgroups corresponding to the above subalgebras can be constructed quite easily. Further, we describe the orbits of these subgroups as they act on ℝ^{2}, dismissing those that have more than 1-dimensional symmetry algebras (i.e., open subsets of straight lines and parabolas). As a result, we obtain the following list of affinely homogeneous curves in the plane with 1-dimensional symmetry algebras:

a) none (i.e., all the orbits have more than 1-dimensional symmetry algebras);

b) none;

c) y = x ln x;

d) y = x^{α} (α ≠ 0, 1, 1/2, 2; α ∼ 1/α);

e) y = ln x;

f) x^{2} + y^{2} = e^{2α arg(x+iy)}.

33 CLASSIFICATION OF HOMOGENEOUS SUBMANIFOLDS Example 2. We shall now describe all homogeneous curves in projective geometry. In this case G = SL(3, ℝ), and the stabilizer G_{0} has the form

G_{0} = \{(\begin{matrix}A & 0 \\ ^{t}B & (det A)^{−1} \end{matrix}) | A ∈ GL(2, ℝ), B ∈ ℝ^{2}\} .

We omit the details of calculations, which present no real difficulties, and provide the final form of the subspaces V_{i} ⊂ 𝔤/𝔤_{i} and subgroups G_{i+1} that arise at each step of the algorithm:

0. V_{0} = \{(\begin{matrix}0 & 0 & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{0}\}, G_{1} = \{(\begin{matrix}a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & 1/a_{11}a_{22} \end{matrix})\};

1. V_{1} = \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{1}\}, G_{2} = \{(\begin{matrix}a_{11} & 0 & 0 \\ a_{21} & 1 & 0 \\ a_{31} & a_{32} & 1/a_{11} \end{matrix})\};

1a. V_{1} = \{(\begin{matrix}0 & 0 & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{1}\}, G_{2} = G_{1};

2. V_{2} = \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{2}\}, G_{3} = \{(\begin{matrix}a_{11} & 0 & 0 \\ a_{11}a_{32} & 1 & 0 \\ a_{31} & a_{32} & 1/a_{11} \end{matrix})\};

3. V_{3} = \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{3}\}, G_{4} = \{(\begin{matrix}a_{11} & 0 & 0 \\ a_{11}a_{32} & 1 & 0 \\ a_{11}a_{32}^{2}/2 & a_{32} & 1/a_{11} \end{matrix})\};

4. V_{4} = \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ ±x & 0 & 0 \end{matrix}) + 𝔤_{4}\}, G_{5} = \{(\begin{matrix}1 & 0 & 0 \\ a_{32} & 1 & 0 \\ a_{32}^{2}/2 & a_{32} & 1 \end{matrix})\};

4a. V_{4} = \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix}) + 𝔤_{4}\}, G_{5} = G_{4};

5. V_{5} = \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ ±x & 0 & 0 \end{matrix}) + 𝔤_{5}\}, G_{6} = \{E_{3}\};

6. 𝔥 = V_{6} = \{(\begin{matrix}0 & x & 0 \\ αx & 0 & x \\ ±x & αx & 0 \end{matrix}) + 𝔤_{5}\}, G_{7} = G_{6}.

1a and 4a are the two cases in which the dimension of the symmetry algebras is greater than 1. Indeed, in the case 1a we have

𝔥 = \{(\begin{matrix}a_{11} & 0 & 0 \\ a_{21} & a_{22} & x \\ a_{31} & a_{32} & −a_{11} − a_{22} \end{matrix})\} .

It can be easily verified that the corresponding homogeneous submanifold is the line.

In the case 4a, the symmetry algebra 𝔥 has the form

𝔥 = \{(\begin{matrix}a_{11} & x & 0 \\ a_{32} & 0 & x \\ 0 & a_{32} & −a_{11} \end{matrix})\} .

Note that this algebras consists precisely of those elements X ∈ 𝔰𝔩(3, ℝ) that preserve the symmetric bilinear form with the matrix B = ( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} ), so that

𝔥 = {X ∈ 𝔰𝔩(3, R) | ^{t}XB + BX = 0}.

It follows that the corresponding subgroup H ⊂ G is the identity component of the subgroup

\overline{H} = \{X ∈ SL(3, ℝ) | ^{t}XBX = B\}.

This immediately implies that the orbit of H passing through the point o = [0 : 0 : 1] is the quadric given by the equation 2x_{1}x_{3} − x_{2}^{2} = 0 in the homogeneous coordinates [x_{1} : x_{2} : x_{3}].

In the general case of the symmetry algebra 𝔥 of a homogeneous submanifold being 1-dimensional, the construction of H and of the explicit equation of the submanifold again reduces to the computation of matrix exponents involving parameters.

Therefore, in order to describe explicitly all homogeneous curves with 1dimensional symmetry algebra, we apply a different method based on the classification of 1-dimensional subalgebras in 𝔰𝔩(3, ℝ). This classification is equivalent to classifying matrices in 𝔰𝔩(3, ℝ) up to conjugation and up to a nonzero constant factor. Using the Jordan forms of 3 × 3, we easily find that any 1-dimensional subalgebra is equivalent to one and only one of the following:

a) \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})\};

b) \{(\begin{matrix}0 & x & 0 \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix})\};

c) \{(\begin{matrix}x & x & 0 \\ 0 & x & 0 \\ 0 & 0 & −2x \end{matrix})\};

d) \{(\begin{matrix}x & 0 & 0 \\ 0 & αx & 0 \\ 0 & 0 & −(1 + α)x \end{matrix})\}, α ∼ \frac{1}{α}, −1 − α, −\frac{1}{α+1}, −\frac{α}{α+1}, −\frac{α+1}{α};

e) \{(\begin{matrix}αx & x & 0 \\ −x & αx & 0 \\ 0 & 0 & −2αx \end{matrix})\}, α ∼ −α.

Constructing the corresponding subgroups H and describing their orbits, we obtain the following homogeneous curves with 1-dimensional symmetry algebra:

a) none;

b) none; c) y = x ln x;

d) y = x^{a}, where a = \frac{2α+1}{α+2} (a ≠ 0, 1, 2, \frac{1}{2}, −1; a ∼ 1 − a, \frac{1}{a}, \frac{1}{1−a}, \frac{a}{a−1}, \frac{a−1}{a});

e) x^{2} + y^{2} = e^{b arg(x+iy)}, where b = 6α (b ≠ 0; b ∼ −b). We remark that the equations of projectively homogeneous curves are writ-

ten in the coordinates of an affine chart in which the curves are affinely homogeneous. Note that the list of the affinely projective curves contains less curves than the similar list of affinely homogeneous curves, because the affinely homogeneous curves y = x ln x and y = ln x are not equivalent with respect to the group of affine transformation, but they can be transformed into each other by means of the projective transformation (x, y) ↦ (1/x, −y/x).

For the sake of convenience, we combine the results of Examples 1 and 2 into a single table in which the canonical equations of both affinely and projectively homogeneous curves are written in affine coordinates. For each curve in the table, we specify the type of geometry (“A” for affine and “P” for projective) to which it belongs. For curves that are affinely homogeneous, but not projectively homogeneous, we provide a projectively homogeneous curve to which they are projectively equivalent.

Table 1. Homogeneous curves in 𝔸^{2} and ℝP^{2}

Equation  Type of Projectively
of curve  geometry equivalent to:
1. y = 0  AP
2. y = x^{2}/2  AP
3. y = x^{a}  AP
4. x^{2} + y^{2} = e^{b arg x+iy}  AP
5. y = ln x  AP
6. y = x ln x  A  5

6. The summary of results

We apply the above-described algorithm to classify locally homogeneous surfaces in the 3-dimensional affine and projective geometries. As in the plane, we distinguish the surfaces with 2-dimensional symmetry algebras from those whose symmetry algebras have dimension ≥ 3. In the first case, we need to classify 2-dimensional subalgebras in the Lie algebras aff(3, 𝕂) = 𝔤𝔩(3, 𝕂)⋌𝕂^{3} and 𝔰𝔩(4, 𝕂) (𝕂 = ℂ, ℝ), respectively. To do this, we use the method described in [15]. Detailed computations are provided in [4] and [2]. Here we only list the results, combining them, as in Examples 1 and 2, into a single table. Thus, every locally homogeneous surface in 3-dimensional real affine (resp., projective) geometry is either an open subset of a cylinder or quadric, or is equivalent to an open subset of one of the surfaces listed in Table 2 having the type AP or A (resp., AP or P).

Remark 1. Every locally homogeneous cylinder is equivalent to a cylinder whose base is a locally homogeneous curve in the plane. For a list of all locally homogeneous plane curves, see Examples 1,2.

Table 2. Homogeneous surfaces in 𝔸^{3} and ℝP^{3}

Equation  Type of Projectively
of surface  geometry equivalent to:
1. z = x^{a}y^{b}  AP
2. z = (x^{2} + y^{2})^{a}e^{b arg(x+iy)}  AP
3. arg(x + iy) = a ln \frac{x^{2}+y^{2}}{1+z^{2}} + b arctg z  P
4. z = ln x + a ln y  AP
5. z = x(a ln x + ln y) A (4, −a − 1)
6. z = a arg(x + iy) + b ln(x^{2} + y^{2})  AP

a = 1, b ≥ 0 or a = 0, b = 1

7. z = y^{2} ± x^{a}  AP
8. xz = y^{2} ± x^{a} A (7, −a + 2)
9. z = xy + x^{a}  AP
10. z = xy + e^{a arctg x}(1 + x^{2})  P
11. (z − xy + x^{3}/3)^{2} = a(y − x^{2}/2)^{3}  AP
12. z = xy + ln x  AP
13. z = xy + x ln x  AP
14. z = xy + x^{2} ln x A 12
15. z = xy + e^{x}  AP
16. z = xy + (1 + x^{2}) arctg x  P
17. z = y^{2} ± ln x  AP
18. xz = y^{2} ± x^{2} ln x A (17, ∓)
19. z = y^{2} ± x ln x  AP
20. xz = y^{2} ± x ln x A (19, ∓)
21. z = y^{2} ± e^{x}  AP

Remark 2. For certain values of parameters, the above surfaces may turn into quadrics or even cylinders. Moreover, in each case there is an equivalence relation on the set of parameters. For more details, see [4] and [2].

Remark 3. The complex version of this theorem is obtained by excluding the surfaces 2, 3, 6, 10, 16 and changing the signs ± to simply + (or −).

On passing to the 4-dimensional affine and projective geometries, we face a number of difficulties as the volume of computations increases and results become rather unwieldy. In this article, we list the results of N. Mozhei [11], who used the methods described above to classify all 3-dimensional homogeneous submanifolds in 𝔸^{4} and ℝP^{4} whose symmetry algebras have dimension ≥ 4, i.e., for which the algebra 𝔤_{∞} is nonzero. In some sense, these submanifolds can be thought of as the most symmetrical of all. In particular, this class contains all cylinders and quadrics. As in the previous cases, we combine all results into a single table. Thus, every hypersurface in 𝔸^{4} (resp., ℝP^{4}) is either an open subset of a cylinder or of a quadric, or is equivalent to an open subset of one of the submanifolds in Table 3 having the type AP or A (resp., AP or P).

37 CLASSIFICATION OF HOMOGENEOUS SUBMANIFOLDS Table 3. Homogeneous surfaces in 𝔸^{4} and ℝP^{4} with symmetry algebra of dimension ≥ 4

Equation  Type of Projectively
of surface  geometry equivalent to:
1. x_{4} = x_{1}x_{3}^{2} + x_{2}x_{3}  AP
2. x_{2}^{2} + x_{1}x_{2}x_{3} + x_{4}x_{1}^{2} = 0 A 1
3. x_{4} = x_{1}x_{3}^{2} + x_{2}x_{3} + x_{3}^{4}  AP
4. x_{4} = x_{1}^{2}x_{3} + x_{2}x_{3}  AP
5. x_{4}x_{3} + x_{1}x_{3}^{2} + x_{2}^{2} = 0 A 4
6. x_{4} = x_{1}^{2} + x_{1}x_{3}^{2} + x_{2}x_{3} + ax_{3}^{4}  AP
7. x_{2}^{2} + x_{1}x_{3}^{2} + x_{1}x_{4} = 0, x_{1} > 0 ∨ x_{1} < 0  AP
8. x_{4}(1 + x_{1}^{2}) = x_{1}(x_{2}^{2} − x_{3}^{2}) + 2x_{2}x_{3}  P
9. x_{1} ± x_{2}^{2} = x_{3}x_{4} + x_{4}^{a}  AP
10. x_{1} ± x_{2}^{2} = x_{3}x_{4} + (1 + x_{4}^{2})e^{a arctg x_{4}}  P
11. x_{1} ± x_{2}^{2} = x_{3}x_{4} + ln x_{4}  AP
12,+ x_{1} + x_{2}^{2} = x_{3}x_{4} + x_{4} ln x_{4}  AP
12,− x_{1} − x_{2}^{2} = x_{3}x_{4} + x_{4} ln x_{4}  A  (12, +)
13. x_{1} ± x_{2}^{2} = x_{3}x_{4} + x_{4}^{2} ln x_{4}  A  (11, ∓)
14. x_{1} ± x_{2}^{2} = x_{3}x_{4} + e^{x_{4}}  AP
15. x_{1} + x_{2}^{2} = x_{3}x_{4} + (1 + x_{4}^{2}) arctg x_{4} = 0  P
16. x_{1}^{2} ± (x_{2}^{2} + x_{3}^{2}) = x_{4}^{a}  AP
17. x_{1}^{2} + x_{2}^{2} − x_{3}^{2} = x_{4}^{a}  AP
18. x_{1}^{2} ± (x_{2}^{2} + x_{3}^{2}) = e^{x_{4}}  AP
19. x_{1}^{2} + x_{2}^{2} − x_{3}^{2} = e^{x_{4}}  AP
20. x_{1}^{2} ± (x_{2}^{2} + x_{3}^{2}) = (1 + x_{4}^{2})e^{a arctg x_{4}}  P
21. x_{1}^{2} + x_{2}^{2} − x_{3}^{2} = (1 + x_{4}^{2})e^{a arctg x_{4}}  P
22. x_{1} ± (x_{2}^{2} + x_{3}^{2}) = x_{4}^{a}  AP
23. x_{1} + x_{2}^{2} − x_{3}^{2} = x_{4}^{a}  AP
24. x_{1}x_{4} ± (x_{2}^{2} + x_{3}^{2}) = x_{4}^{a}  A  (22, −a + 2)
25. x_{1}x_{4} + x_{2}^{2} − x_{3}^{2} = x_{4}^{a}  A  (23, −a + 2)
26. x_{1} ± (x_{2}^{2} + x_{3}^{2}) = ln x_{4}  AP
27. x_{1} + x_{2}^{2} − x_{3}^{2} = ln x_{4}  AP
28. x_{1}x_{4} ± (x_{2}^{2} + x_{3}^{2}) = x_{4}^{2} ln x_{4}  A  (26, ∓)
29. x_{1}x_{4} + x_{2}^{2} − x_{3}^{2} = x_{4}^{2} ln x_{4}  A  27
30. x_{1} ± (x_{2}^{2} + x_{3}^{2}) = x_{4} ln x_{4}  AP
31. x_{1} + x_{2}^{2} − x_{3}^{2} = x_{4} ln x_{4}  AP
32. x_{1}x_{4} ± (x_{2}^{2} + x_{3}^{2}) = x_{4} ln x_{4}  A  (30, ∓)
33. x_{1}x_{4} + x_{2}^{2} − x_{3}^{2} = x_{4} ln x_{4}  A  31
34. x_{1} ± (x_{2}^{2} + x_{3}^{2}) = e^{x_{4}}  AP
35. x_{1} + x_{2}^{2} − x_{3}^{2} = e^{x_{4}}  AP
36. (x_{1}x_{3}^{2} − x_{4}x_{2}x_{3} + x_{4}^{3}/3)^{2} = −8/9(x_{3}x_{2} − x_{4}^{2}/2)^{3}  A  cylinder
37. x_{1}x_{2}x_{3} + x_{1}^{2}x_{4} + x_{2}^{3} = 0  A  cylinder

B. Doubrov and B. Komrakov 38

Remark 4. In the case 7, the surface consists of two connected components (determined by the conditions x_{1} > 0 and x_{1} < 0) that are not projectively (and hence, affinely) equivalent to each other.

References

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[4] B. Doubrov, B. Komrakov, and M. Rabinovich, Homogeneous surfaces in threedimensional affine geometry, Preprint Univ. Oslo, no. 41, 1995.

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International Sophus Lie Centre E-mail address: Doubrov@islc.minsk.by, Boris.Komrakov@wis.kuleuven.ac.be

Received July 27, 1999