SMOOTHNESS CONDITIONS IN COHOMOGENEITY ONE MANIFOLDS

We present an efficient method for determining the conditions that a metric on a cohomogeneity one manifold, defined in terms of functions on the regular part, needs to satisfy in order to extend smoothly to the singular orbit.


Introduction
A group action is called a cohomogeneity one action if its generic orbits are hypersurfaces. Such actions have been used frequently to construct examples of various types: Einstein metrics, soliton metrics, metrics with positive or nonnegative curvature and metrics with special holonomy. See [4], [6], [7], [8], [12] for a selection of such results. The advantage of such a metric is that geometric problems are reduced to studying its behavior along a fixed geodesic c(t) normal to all orbits. The metric is described by a finite collection of functions of t, which for each time specifies the homogeneous metric on the principal orbits. One aspect one needs to understand is what conditions these functions must satisfy if regular orbits collapse to a lower-dimensional singular orbit. These smoothness conditions are often crucial ingredients in obstructions, e.g., to non-negative or positive curvature, see [9], [14], [15]. The goal of this paper is to devise a simple procedure in order to derive such conditions explicitly.
The local structure of a cohomogeneity one manifold near a collapsing orbit can be described in terms of Lie subgroups H ⊂ K ⊂ G with K/H = S , > 0. The action of K on S extends to a linear action on D = D +1 ⊂ R +1 and thus M = G × K D is a homogeneous disc bundle, where K acts as (g, p) → (gk −1 , kp), and with boundary G× K ∂ D = G× K K/H = G/H a principal orbit. The Lie group G acts by cohomogeneity one on M by left multiplication in the first coordinate. A compact (simply connected) cohomogeneity one manifold is the union of two such homogeneous disc bundles. For simplicity we write M = G × K V with V R n . Given a smooth G invariant metric on the open dense set of regular points, i.e., the complement of the lower-dimensional singular orbit, the problem is when the extension of this metric to the singular orbit is smooth. We first simplify the problem as follows: Theorem 1. Let G act by cohomogeneity one on M = G × K V and g be a smooth cohomogeneity one metric defined on the set of regular points in M . Then g has a smooth extension to the singular orbit if and only if it is smooth when restricted to every 2-plane in the slice V containingċ(0).
As we will see, it follows from the classification of transitive actions on spheres, that it is sufficient to require the condition only for a finite set of 2-planes P i = {ċ(0), v i }, one for each irreducible summand in the isotropy representation of the sphere K/H. Thus at most four 2-planes are necessary. Furthermore, L i = exp(θv i ) ⊂ K is a closed one-parameter group and hence the action of L on V and on a K invariant complement of k in g splits into 2-dimensional invariant subspaces i isomorphic to C, on which L acts by multiplication with e iniθ . The integers n i are determined by the weights of the representation of K on V and the tangent space of G/K. These integers will determine the smoothness conditions, see Tables  B and C. To be more explicit, choose a normal geodesic c : [0, ∞) → V orthogonal to all orbits. The metric on the regular part is determined by its values along c, and via the action of G this determines the metric on M . Denote by g, h the Lie algebras of G and H, and let n be an Ad H invariant complement of h ⊂ g. Since the stabilizer group along c is constant equal to H, n can be identified with the tangent space to the regular orbits along c using action fields, i.e., X ∈ n → X * (c(t)). Thus g = dt 2 + h t , where h t , t > 0 is a family of G-invariant metrics on the regular orbits g · c(t) = G/H, depending smoothly on t. Equivalently, h t is a smooth family of Ad H invariant inner products on n.
The metric is described in terms of the length of Killing vector fields. We choose a basis X i of n and let X * i be the corresponding Killing vector fields. Then X * i (c(t)) is a basis ofċ ⊥ (t) ⊂ T c(t) M for all t > 0 and the metric is determined by the r functions g ij (t) = g(X * i , X * j ) c(t) , i ≤ j. Combining the finite set of smoothness conditions obtained from Theorem A, we will show that: Theorem 2. Let g ij (t), t > 0 be a smooth family of positive definite matrices describing the cohomogeneity one metric on the regular part along a normal geodesic c(t). Then there exist integers a k ij and d k , with d k ≥ 0, such that the metric has a smooth extension to all of M if and only if i,j a k ij g ij (t) = t d k φ k (t 2 ) for k = 1, . . . , r, and t > 0 where φ 1 , . . . , φ r are smooth functions defined for t ≥ 0.
We will show that this system of r equations can also be solved for the coefficients g ij of the metric. The integers a k ij are determined by the Lie brackets [X i , X j ], and d k by the integers n i . These equations hold for all t in the case of a complete metric on a non-compact manifold, and on the complement of the second singular orbit when the manifold is compact. We will illustrate in some specific examples that it is straightforward to determine these integers.
The problem of smoothness was studied in [5] as well. There it was shown that smoothness is equivalent to showing that the k-th order Taylor polynomial of g ij is the restriction of an Ad H invariant homogeneous polynomial of degree k in dim V variables with values in S 2 n. In practice this description is difficult to apply, since one needs explicit expressions for these polynomials.
In two future papers, we will show that our new description is useful in proving general theorems about cohomogeneity one manifolds. In [16] we classify curvature homogeneous cohomogeneity one metrics in dimension 4, where the smoothness conditions at the singular orbit make the problem algebraically tractable. In [17] we solve the initial value problem, starting at the singular orbit, for Einstein metrics, soliton metrics or for prescribing the Ricci tensor. The equations can be described in terms of the smooth functions φ i , and the system is smooth if and only if the values φ i (0) satisfy certain compatibility conditions. These can be solved for some of the values φ i (0), and the remaining ones are free parameters. For this it is also important to understand the smoothness conditions for a symmetric 2 tensor (in particular the Ricci tensor), which we indicate in Section 3.4. The initial value problem for Einstein metrics was solved in [5], only under strong assumptions on the adjoint representation of H on n using different more complicated methods.
The paper is organized as follows. After discussing some preliminaries in Section 1, we prove Theorem A in Section 2. In Section 3 we describe how the action of the one-parameter group L ⊂ K on V and on the tangent space to the singular orbit is used to derive the smoothness conditions. This is an over determined system of equations, and we will show how it can be reduced to the system in Theorem B. In Section 4 we illustrate the method in some specific examples. There the reader will also find step by step instructions of how the process works. In order to facilitate the procedure we determine the integers d k for the action of K on V in Section 5.

Preliminaries
For a general reference for this Section see, e.g., [1], [2]. A noncompact cohomogeneity one manifold is given by a homogeneous vector bundle and a compact one by the union of two homogeneous disc bundles. Since we are only interested in the smoothness conditions near a singular orbit, we restrict ourselves to only one such bundle. Let H, K, G be Lie groups with inclusions H ⊂ K ⊂ G such that H, K are compact and K/H = S . The transitive action of K on S extends (up to conjugacy) to a unique linear action on the disc V = R +1 . We can thus define the homogeneous vector bundle M = G × K V and G acts on M via left action in the first component. This action has principal isotropy group H, and singular isotropy group K at a fixed base point p 0 ∈ G/K contained in the singular orbit. A disc D ⊂ V can be viewed as the slice of the G action since, via the exponential map, it can be identified G-equivariantly with a submanifold of M orthogonal to the singular orbit at p 0 .
Given a G-invariant metric g on the regular part of the G action, i.e., on the complement of G · p 0 , we want to determine when the metric can be extended smoothly to the singular orbit. We choose a geodesic c parameterized by arc length and normal to all orbits with c(0) = p 0 . Thus, with the above identification, c(t) ⊂ V . At the regular points c(t), i.e., t > 0, the isotropy is constant equal to H. We fix an Ad H invariant splitting g = h ⊕ n and identify the tangent space T c(t) G/H =ċ ⊥ ⊂ T c(t) M , with n via action fields: X ∈ n → X * (c(t)). H acts on n via the adjoint representation and a G invariant metric on G/H is described by an Ad H invariant inner product on n. For t > 0 the metric along c is thus given by g = dt 2 + h t with h t a one-parameter family of Ad H invariant inner products on the vector space n, depending smoothly on t. Conversely, given such a family of inner products h t , we define the metric on the regular part of M by using the action of G.
By the slice theorem, for the metric on M to be smooth, it is sufficient that the restriction to the slice V is smooth. This restriction can be regarded as a map g(t) : V → S 2 (n). The metric is defined and smooth on V \ {0}, and we need to determine when it admits a smooth extension to V .
We choose an Ad H invariant splitting where Ad H acts trivially on n 0 and irreducibly on n i for i > 0. On n i , i > 0 the inner product h t is uniquely determined up to a multiple, whereas on n 0 it is arbitrary. Furthermore, n i and n j are orthogonal if the representations of Ad H are inequivalent. If they are equivalent, inner products are described by 1, 2 or 4 functions, depending on whether the equivalent representations are real, complex or quaternionic. Next, we choose a basis X i of n, adapted to the above decomposition, and thus the metrics h t are described by a collection of smooth functions g ij (t) = g(X * i (c(t)), X * j (c(t))), t > 0. In order to be able to extend this metric smoothly to the singular orbit, they must satisfy certain smoothness conditions at t = 0, which we will discuss in the next two Sections. Notice that in order for the metric to be well defined on M, the limit of h t , as t → 0, must exist and be Ad K invariant at the singular orbit.
Choosing an Ad K invariant complement to k ⊂ g, we obtain the decompositions where we can also assume that n i ⊂ p or n i ⊂ m. Here m can be viewed as the tangent space to the singular orbit G/K at p 0 = c(0) and p as the tangent space of the sphere K/H ⊂ V . It is important for us to identify V in terms of action fields. For this we send X ∈ p toX := lim t→0 X * (c(t))/t ∈ V . Since K preserves the slice V and acts linearly on it, we thus have X * (c(t)) = tX ∈ V . In this language, V ċ(0) ⊕ p. For simplicity we denoteX again by X and, depending on the context, use the same letter if considered as an element of p or of V .
Notice that since K acts irreducibly on V , an invariant inner product on V is determined uniquely up to a multiple. Since for any G invariant metric we fix a geodesic c, which we assume is parameterized by arc length, this determines the inner product on V , which we denote by g 0 . Thus g 0 = g c(0) | V for any G invariant metric for which c is a normal geodesic.
K acts via the isotropy action Ad(K)| m of G/K on m and via the slice representation on V . The action on V is determined by the fact that K/H = S . Notice though that the action of K on S , and hence on V is often highly ineffective. If R ⊂ K is the ineffective kernel of the action, then there exists a normal subgroup N ⊂ K with K = (R × N )/Γ where Γ is a finite subgroup of the center of R × N . Thus N acts almost effectively and transitively on S with stabilizer group N ∩ H. We list the almost effective actions by connected Lie groups acting transitively on spheres in Table A. From this, one can recover the action of K on V simply from the embedding H ⊂ K.
The smoothness conditions only depend on the Id component of K since, as we will see, they are determined by certain one-parameter groups L S 1 ⊂ K 0 . Since also L ⊂ N , the smoothness conditions only depend on the Id component of N as well.
We finally collect some specific properties of transitive actions on spheres.
Lemma 3. Let S = K/H ⊂ V be a sphere, with K acting almost effectively and H the stabilizer group of v 0 ∈ V . If k = h ⊕ p is an Ad H invariant decomposition, we have: (a) If p 1 ⊂ p is an Ad H irreducible summand with dim p 1 > 1, then H acts transitively on the unit sphere in p 1 , (b) If p i ⊂ p, i = 1, 2, are two Ad H irreducible summands with dim p i > 1 and X 1 , Y 1 ∈ p 1 and X 2 , Y 2 ∈ p 2 two pairs of unit vectors, then there exists an h ∈ H such that Ad(h)X i = Y i . (c) If X ∈ p lies in an Ad H irreducible summand, or a trivial one, then exp(tX) is a closed one-parameter group in K and leaves invariant the 2-plane spanned by v 0 and X * (v 0 ).
Proof. Part (a) can be verified for each sphere separately, using the description of the adjoint representation, see, e.g., [18]. Part (b) is easily verified in case 5, 5 and 6, 6 in Table A. In the remaining case of K = Spin (9) and H = Spin(7) we have p = p 1 ⊕ p 2 with Spin(7) acting on p 1 R 7 via the 2-fold cover Spin(7) → SO(7), and on p 2 R 8 via its spin representation. We can first choose an h ∈ H with Ad(h)(X 1 ) = Y 1 . The claim then follows since the stabilizer of H at Y 1 ∈ R 7 is Spin (6), and the restriction of the spin representation of Spin(7) on R 8 to this stabilizer is the action of Spin(6) = SU(4) on C 4 , which is transitive on the unit sphere.
Since exp(tX) is the flow of the action field X * , part (c) is equivalent to saying that exp(tX) · v 0 is a great circle in S . Recall that for a normal homogeneous metric, i.e., a metric on K/H induced by a biinvariant metric on K, the geodesics are of the form exp(tX) · v 0 for some X ∈ p. This implies the claim if Ad H acts irreducibly on p. In all other cases, one can view the irreducible summand as the vertical or horizontal space of a Hopf fibration. The round metric on S is obtained from the metric induced by a biinvariant metric on K by scaling the fiber, see [10], Lemma 2.4. But such a change does not change the geodesics whose initial vector is vertical or horizontal. By part (a), the one-parameter groups exp(tX) are either all closed in K, or none of them are. But for each transitive sphere one easily finds one vector v where it is closed, see Section 6.

Reduction to a 2-plane
In this Section we show how to reduce the question of smoothness of the metric on M = G × K V to a simpler one. If dim V = 1, i.e., the orbit G/K is exceptional, smoothness (of order C k or C ∞ ) of the metric is equivalent to the invariance with respect to the Weyl group since the slice is the normal geodesic. Recall that the Weyl group element is an element w ∈ K such that w(ċ(0) = −ċ(0), and is hence uniquely determined mod H. Hence we only need to discuss the conditions at singular points. i.e., dim V > 1.
At a singular point, the slice theorem for the action of G implies that the metric is smooth if and only if its restriction to a slice V , i.e., g| V : V → S 2 (p ⊕ m) is smooth. Indeed, in a neighborhood W of the slice we have an equivariant diffeomorphism U × V → W : (x, p) → exp(x)p, where U is a sufficiently small neighborhood of 0 ∈ n. We choose for each Ad H irreducible summand in p an (arbitrary) vector v i = 0. If there exists a 3-dimensional trivial module p 0 ⊂ p, we pick in p 0 an arbitrary fixed basis. Proof. First notice that by Lemma 3(a), and since the metric is fixed along the normal geodesic c, the assumption implies that the metric is smooth when restricted to a 2-plane spanned byċ(0) and v, where v is any vector in an irreducible p module.
It is sufficient to show that g(X, Y )| V is smooth for any non-vanishing smooth vector fields X, Y defined on V , i.e., X, Y : V → T M . We will use equivariance of the metric with respect to the action of K on V . i.e., for the metric g as well as all of its derivatives.
We first define the metric at 0 ∈ V and show it is K invariant, as required. For this, define g(X, Y )(0) = lim t→0 g(X, Y )(c(t)). If P i is spanned byċ(0) and v i , then by Lemma 3 (c) the one-parameter group L = exp(tv i ) preserves the plane P i , and equivariance with respect to L ⊂ K implies that g(0) is invariant under L. By Lemma 3 (a), the same is true for exp(tv) for any vector v lying in an Ad H irreducible submodule of p. But such one-parameter groups, together with H, generate all of K. Indeed, this follows from the fact that d/dt| t=0 (exp(tv) exp(tw)) = [v, w] and that h ⊕ [p, p] is an ideal in g.
We next prove continuity. Let p i be a sequence of points p i ∈ V \ {0} such that p i → 0. We want to show that g(X, Y )(p i ) converges to g(X, Y )(0). For this, let w 0 be an accumulation point of w i = p i /|p i | and choose a subsequence w i → w 0 . Since K acts transitively on a sphere in V , we can then choose r i ∈ K such that r i w i = w 0 and r i → e ∈ K, as well as k 0 ∈ K with k 0 w 0 =ċ(0). Setting k i = k 0 r i , it follows that k i w i =ċ(0) with k i → k 0 , which implies that k i p i lies on the geodesic c. Hence equivariance of the metric, and continuity of the metric along the normal geodesic, implies that where we also used that the metric at the origin is invariant under K. Since the same argument holds for any accumulation point of the sequence w i , this proves continuity.
Next, we prove the metric is C 1 . For simplicity we first assume that the action of H on p is irreducible and non-trivial and hence H acts transitively on the unit sphere in p. By assumption, the metric is smooth when restricted to the 2-plane P spanned by v ∈ p andċ(0). Given a vector w ∈ V , possibly w =ċ(0), we need to show that the derivative with respect to w extends continuously across the origin, i.e., that for any sequence p i ∈ V with p i → 0. Let us first show that the right-hand side derivative in fact exists. For this, since K acts transitively on every sphere in V , we can choose k ∈ K such that kw ∈ P and hence: where we have used K equivariance away from the origin and K invariance of g at the origin. But the right side is the derivative which exists by assumption since kw ∈ P . Now choose as before k i ∈ K such that k i p i lies on the geodesic c. Since H acts transitively on the unit sphere in p, and since p is the orthogonal complement toċ(0) ∈ V , we can choose h i ∈ H such that h i k i w lies in P . As before, we can assume that k i → k 0 and h i → h 0 . Equivariance and smoothness of the metric away from the origin implies that for each fixed i Thus the metric is C 1 . The proof proceeds by induction. Assume the metric is C k . This means that T (w 1 , . . . , w k , X, Y )(p) = ∂ k ∂w1...∂w k g(X, Y )(p) is a smooth multi linear form on the slice V which is equivariant in all its arguments. We can thus use the same proof as above to show that extends continuously across the origin, and hence the metric is C k+1 . We now extend the above argument to the case where p is not irreducible. Let P i be the 2-plane spanned by v i andċ(0). We first observe that any vector in p, can be transformed by the action of H into a linear combination of the vectors v i . Indeed, if we look at the possible isotropy actions of K/H in Table A, one sees that besides the trivial module (in which we chose a basis) there are at most two non-trivial modules and Lemma 3(b) implies the claim. Following the strategy in the previous case, we choose k i ∈ K such that k i p i lies on the geodesic c, and By linearity of the derivative, and since the metric is smooth on P i by assumption, we have The proof now continues as before.
Remark 1. Notice that unless the group K is Sp(n) or Sp(n)·U(1), only one or two 2-planes are required. For the exceptions one needs four resp three 2-planes. Notice also, that we can choose any vector v in an irreducible submodule in p. Indeed, the condition is clearly independent of such a choice since H acts transitively on the unit sphere in every irreducible submodule. We point out that Proposition 4 also holds for any tensor on M invariant under the action of G, using the same strategy of proof.

Smoothness on 2-planes
In this Section we show that smoothness on 2-planes can be determined explicitly in a simple fashion.
Recall that on V we have the inner product g 0 with g 0 = g c(0) | V for any G invariant metric with normal geodesic c. We fix a basis e 0 , e 1 , . . . , e k of V , orthonormal in g 0 , such that c is given by the line c(t) = te 0 = (t, 0, . . . , 0). The tangent space to M at the points c(t), t > 0 can be identified withċ(t) ⊕ m ⊕ p via action fields. The metric g = dt 2 + h t on the set of regular points in M is determined by a family of Ad H invariant inner products h t on m ⊕ p, t > 0, which depend smoothly on t. Furthermore, m and p are orthogonal at t = 0, but not necessarily for t > 0. The inner products h t extend in a unique and smooth way to V \ {0} via the action of K. In order to prove smoothness at the origin, it is sufficient to show that g(X i , X j ) is smooth for some smooth vector fields which are a basis at every point in a neighborhood of c(0). For this we use the action fields X * i corresponding to an appropriately chosen basis X i of m, restricted to the slice V , and the (constant) vector fields e i on V . Recall also that we identify p with a subspace of V by sending X ∈ p to lim t→0 X * (c(t))/t ∈ V and that X * (c(t)) = tX. Finally, we have the splitting p = p 1 ⊕ · · · ⊕ p s into Ad H irreducible subspaces.
According to Proposition 4, it is sufficient to determine smoothness on a finite list of 2-planes. Let P * ⊂ V be one of those 2-planes, spanned by e 0 =ċ(0) and X ∈ p i for some i. We normalize X such that L := {exp(θX) | θ ∈ R, 0 ≤ θ ≤ 2π} is a closed one-parameter subgroup of K. By Lemma 3, the one-parameter group L preserves P * , but may not act effectively on it, even if K acts effectively on V . Since L S 1 , acting via rotation on P * , the ineffective kernel is L ∩ H. Let a be the order of the finite cyclic group L ∩ H. Equivalently, a is the largest integer with exp (2π/a)X c(0) = c(0), or equivalently exp (2π/a)X ∈ H. Thus X/a has unit length in g 0 and L operates on P * as a rotation R(aθ) in the orthonormal basiṡ c(0), X/a. We can also assume a > 0 by replacing, if necessary, X by −X. This integer a will be a crucial ingredient in the smoothness conditions. Notice that a is the same for any vector X ∈ p i and we can thus simply denote it by a i . In the Appendix we will compute the integers a i for each almost effective transitive action on a sphere.
The action of L on m decomposes m: for some integers d i . Similarly we have a decomposition of V : . We choose the basis e i of V and X i of m such that it is adapted to this decomposition and oriented in such a way that a, d i and d i are positive. For simplicity, we denote the basis of i by Y 1 , Y 2 , the basis of i by Z 1 , Z 2 , and reserve the letter X for the one-parameter group L = exp(θX). We choose the vectors Z i ∈ p such that they correspond to e i+1 under the identification p ⊂ V and hence Z * i (c(t)) = te i+1 ∈ V , as well as X * (c(t)) = te 0 . We determine the smoothness of inner products module by module, and observe that an L invariant function f on P * extends smoothly to the origin if and only if its restriction to the line te 0 is even, i.e., f (te 0 ) = g(t 2 ) with g : (− , ) → R smooth. Furthermore, we use the fact that the metric V → S 2 (p ⊕ m) is equivariant with respect to the action of K, and hence L. Once the condition is determined when inner products are smooth when restricted to P * , we restrict to the geodesic c to obtain the smoothness condition for h t .
In the following, φ i (t) stands for a generic smooth function defined on an interval (− , ).
We will separate the problem into three parts: smoothness of scalar products of elements in m, in p and mixed scalar products between elements of m and p. We will start with the easier case of the metric on p.

Smoothness on p
Recall that on a 2-plane a metric given in polar coordinates by dt 2 + f 2 (t) dθ 2 is smooth if and only if f extends to a smooth odd function with f (0) = 0 and f (0) = 1, see, e.g., [11]. If X has unit length in the Euclidean metric g 0 , we have X * = ∂/∂θ in the 2-plane spanned byċ(0) and X. Hence smoothness on p is equivalent to: for some smooth function φ, defined on an interval (− , ). Notice that p i and p j , for i = j, are orthogonal for any G invariant metric, unless (K, H) = (Sp(n), Sp(n − 1)), in which case there exists a 3-dimensional module p 0 on which Ad H acts as Id. We choose three vectors X i ∈ p 0 , orthonormal in g 0 .
It may sometimes be more convenient, as we do in the proofs, to normalize X such that L = {exp(θX) | 0 ≤ θ ≤ 2π} is a closed one-parameter group in K. In that case, let t 0 be the first value such that exp(t 0 X) ∈ H. Then t 0 = 2π/a for a = |L ∩ H| and hence X/a has unit length in g 0 .
Thus in this normalization we need to replace (2) by: For a 3-dimensional module p 0 we will see in Section 5 that a i = 1 and hence in this case (3) remains valid. See [13] for a more detailed description.
Remark 2. One easily modifies the smoothness conditions if the geodesic is not necessarily parameterized by arc length, but still orthogonal to the regular orbits.
The only difference is that in this case g c(t) (ċ,ċ) = ψ(t)t 2 and g c(t) (X * , X * ) = φ(t)t 2 for X ∈ p with φ, ψ even and φ(0) = ψ(0) > 0, where X has unit length in g 0 . In the second normalization of X we need that

Inner products in m
In the remaining sections L = {exp(θX) | 0 ≤ θ ≤ 2π} is a one-parameter group acting via R(aθ) on −1 . We first describe the inner products in a fixed module i .

Lemma 5.
Let be an irreducible L module in m on which L acts via a rotation R(dθ) in a basis Y 1 , Y 2 . If the metric on is given by for some smooth functions φ k , k = 1, 2, 3.
Proof. The metric on , restricted to the plane P * ⊂ V , can be represented by a matrix G(p) whose entries are functions of p ∈ P * . We identify C and P * C such that the action of L is given by multiplication with e idθ on and e iaθ on P * . The metric G must be L equivariant, i.e., The right-hand side can also be seen as a linear action of L on S 2 R 3 and we may describe it in terms of its (complex) eigenvalues and eigenvectors. We then get: The first equality just reflects the fact that the trace is a similarity invariant. Let w(p) = (g 12 + i(g 11 − g 22 ))(p).
Then the second equality says that w(e iaθ p) = e 2idθ w(p), and the third one is the conjugate of the second. Setting p = te 0 , t ∈ R and replacing θ by θ/a, we get w(e iθ t) = e (2id/a)θ w(t) = (te iθ ) 2d/a t −2d/a w(t).
If we let z = te iθ , then The first equation says that if w(z) is smooth, then w(z) must have a zero of order 2d/a at z = 0. If so, the second equation says that the function z −2d/a w(z) is L-invariant. This means that g 11 + g 22 and z −2d/a w(z) must be smooth functions of |z| 2 . If we restrict z −2d/a w(z) to the real axis and we separate the real and the imaginary part this is equivalent to the existence of smooth functions φ i such that Conversely, given 3 functions g 11 , g 22 , g 12 along the real axis that verify these relations, they admit a (unique) smooth L-invariant extension to C. Indeed, the first two equalities guarantee that z −2d/a w(z) and hence w(z) is a smooth function on P * . The third equality guarantees that g 11 + g 22 , and hence G(p), has a smooth extension to P * .
Remark 3. If a does not divide 2d, the proof shows that w(z) is smooth only if w(t) = 0 for all t. But then g 12 = 0 and g 11 = g 22 is an even function. Thus in this Lemma, as well as in all following Lemmas, in case of a fractional exponent of t, the term should be set to be 0. In practice, this will follow already from Ad H invariance.
Notice also that a Weyl group element is given by w = exp(i d a π). Thus if q = 2d a is odd, w rotates the 2-plane and hence this module is not changed when it is necessary to select another one-parameter group L.
For inner products between different modules we have: Lemma 6. Let 1 and 2 be two irreducible L modules in m with basis Y 1 , Y 2 resp. Z 1 , Z 2 on which L acts via a rotation R(d i θ)with d i > 0. If the inner products between 1 and 2 are given by for some smooth functions φ k .
For inner products with elements in 0 we have: Lemma 7. Let 0 ⊂ m be the module on which L acts as Id, and an irreducible L module with basis Y 1 , Y 2 on which L acts via a rotation R(dθ).
(a) If Y ∈ 0 , then g c(t) (Y * , Y * ) is an even function of t.
for some smooth functions φ k .
Proof. If Y ∈ 0 , then g(Y * , Y * ) is invariant under L and hence an even function.
In case (b), we consider the restriction of the metric to the three-dimensional space spanned by and Y . This can be represented by a matrix whose entries are functions of p ∈ P * . In particular, h i = g(Y * i , Y * ). The action of L on G(p) is given by conjugation with diag(R(dθ), 1). Decomposing into eigenvectors, we get, in addition to the eigenvectors already described in Lemma 5, the eigenvector w(z) = h 1 (z) + ih 2 (z) with eigenvalue e diθ . But w(e iaθ p) = e diθ w(p) implies that z −d/a w(z) is an invariant function. Thus smoothness for the h i functions is equivalent to for some smooth functions φ i .

Inner products between p and m
Recall that for an appropriately chosen basis e 0 , . . . , e k of V , we need to show that the inner products g(e i , X * j ), where X i is a basis of m, are smooth functions when restricted to the plane P * ⊂ V . When restricting to the geodesic c, we obtain the smoothness conditions on the corresponding entries in the metric.
Proof. For part (a) the proof is similar to Lemma 7. On the 3-space spanned by e 0 =ċ(0), e 1 = X, e 2 = Y , the one-parameter group L acts via conjugation with diag(R(aθ), 1) and, using the fact that Y * is orthogonal toċ, the metric is given by with h = g(e 1 , Y * ) and f = g(Y * , Y * ). We already saw that f is an even function, and as in the proof of Lemma 7, we see, when restricted to the geodesic, For part (b) the proof is similar to Lemma 6. On the 4-dimensional space spanned by e 0 , e 1 and Y 1 , Y 2 the group L acts via conjugation by diag(R(aθ, R(dθ)) and the metric is given by . As in the proof of Lemma 6 it follows that h 2 (t) = t |d−a|/a φ 1 (t 2 ) and h 2 (t) = t |d+a|/a φ 2 (t 2 ) and hence h 2 (t) = t d/a+1 φ(t 2 ), and similarly for h . Next the inner products with 0 .
Lemma 9. For Z ∈ 0 we have: for some smooth functions φ i .
Proof. For part (a), let Z = e 1 . Then g(e 1 , Y * ) is L invariant and hence even. Furthermore, it vanishes at t = 0 since the slice is orthogonal to the singular orbit at c(0). Hence g(e 1 , Y * ) = t 2 φ(t 2 ), which implies g c(t) (Z * , Y * ) = t t 2 φ(t 2 ). Similarly for (b), using the proof of Lemma 7, it follows that g c(t) (e 1 , Y * k ) = t di/a φ k (t 2 ). Since d i , a > 0, this already vanishes as required. The proof now finishes as before.
And finally the remaining inner products: Lemma 10. Let i and j with i, j > 0 be two irreducible L modules with basis Z 1 , Proof. (a) We repeat the proof of Lemma 6 for the basis e 1 = Z 1 , e 2 = Z 2 , e 3 = Y 1 , e 4 = Y 2 of i ⊕ j . But if d i = d j , we have to require in addition that the inner products vanish at t = 0, i.e., φ 1 (0) = φ 2 (0) = 0, which means the first matrix must be multiplied by t 2 . The proof then proceeds as before.
(b) We proceed as in Lemma 7(b).
This finishes the discussion of all possible inner products in n = p ⊕ m.

Smoothness conditions for symmetric 2 × 2 tensors
The above methods can be applied to obtain the smoothness conditions for any G invariant tensor, defined along a curve c transverse to all orbits. One needs to take care though, since for a metric g the slice and singular orbit are orthogonal at t = 0, whereas for a general tensor this may not be the case. For the purpose of applying this to the Ricci tensor, we briefly discuss how to derive the smoothness conditions for any symmetric 2 × 2 tensor T . The proofs in Section 3.2 show that for the functions T (m, m) the conditions for T and a metric g are the same.
For T (p, p) the only difference is that now T (X * i , X * j ) = φ 0 t 2 δ ij + φ ij (t 2 )t 4 for X i ∈ p, where X i has unit length in g 0 and φ 0 is a real number, which is allowed to be 0. Notice also that T (p i , p j ) = 0 for 0 < i < j since the Ad H representations are inequivalent.
For the case of T (ċ, m 0 ), as well as T (p, m), one needs to examine the proof of the Lemma's in Section 3.3, keeping in mind that the values of T on the 2plane −1 = {ċ, X} are now more generally given by (5). In some cases, for a metric tensor, certain components are forced to have a zero of two orders higher at t = 0 than a generic symmetric tensor since the regular orbits are orthogonal to the geodesic c. One easily sees that the conditions in Lemma 8(a), Lemma 9(b) and Lemma 10(b) are the same, whereas in Lemma 8(b), Lemma 9(a) and in Lemma 10(a) when d i = d j , the allowed order for T is two less. We summarize the results in Table D. This difference is important when studying Einstein metrics, or prescribing the Ricci tensor, see [17].

Examples
Before we illustrate the method with some examples, let us make some general comments.
We can choose an inner product Q on g which is Ad K invariant on m, equal to g 0 on p under the inclusion p ⊂ V , and such that the decomposition n = p ⊕ m is orthogonal.
If G is compact, one often starts with a biinvariant metric Q on g. We point out though, that then Q| p is not always a multiple of the metric g 0 . Thus one needs to determine the real numbers r i > 0 such that Q| pi = r i g 0 , i = 1, . . . , s, which needs to be used in order to translate the conditions in (2) into a basis of p orthonormal in Q. We point out that if s > 1, r i depends on i since in that case the biinvariant metric Q| K does not restrict to a constant curvature metric on K/H. See Table  2.5 in [10] for the values of r i .
Smoothness is determined by the one-parameter groups L = {exp(θv) | 0 ≤ θ ≤ 2π}, one for each irreducible p module. Since the action of L on m is given by the restriction of Ad K , the exponents d i can be determined in terms of Lie brackets, i.e., on i we have where Y 1 , Y 2 ∈ i are Q orthogonal vectors of the same length. This also determines the orientation of the basis so that d i > 0. The decomposition under L can be recovered from the weight space decomposition of the action of K on m with respect to a maximal abelian subalgebra containing v. Thus, on each irreducible K module in m, we have d i = α i (v), for all weights α i , and hence the largest integer is λ(v) where λ is the dominant weight. The slopes d i are not determined by Lie brackets. One needs to use the knowledge of the embedding H ⊂ K to determine the action of K, and hence L, on V . For the almost effective actions of K on spheres, a choice of the vectors v and the values of a and d i will be described in Section 6.
The functions g ij (t) determining the metric are usually given in terms of a decomposition of h ⊥ = n into Ad H irreducible modules. But the decomposition of m into irreducible modules under L i are usually quite different. Thus the entries of the metric in the Lemmas of Section 3 are linear combinations of g ij . Furthermore, for different 2-planes P * i , the decomposition under L i = exp(θv i ) is again typically not the same since the vectors v i do not lie in a common maximal torus. One may thus obtain different smoothness conditions for different one-parameter groups L i which need to be combined to obtain the full smoothness conditions.
One can now row reduce these equations, which gives rise to relationships between the even functions. Substituting these, one can then express the k metric coefficients in terms of k even functions.
The conditions of order 0 are equivalent to K invariance. The conditions of order 1 are equivalent to equivariance of the second fundamental form B : S 2 T → T ⊥ = V of the singular orbit G/K with tangent space T = T p0 K/H under the action of K. Recall also that one has a Weyl group element w ∈ K with w(ċ(0)) = −ċ(0), uniquely determined mod H. Clearly w ∈ L i for all i, in fact w = exp( π a v i ) up to a change by an element in Ad H . The property of the length squared being even or odd functions is already determined by the action of the Weyl group element on m, see Remark 3 and Section 5.
Summarizing the method one needs to use the following steps: (a) Decompose n into Ad H irreducible modules, which determines the coefficients g ij of the metric. (b) Choose one-parameter groups L = exp(tX), one for each irreducible p module. See Section 6 for convenient choices, as well as the value of the integers a and d i .
(c) Decompose m into the sum of 2-dimensional modules i under the action of L and determine the integers d i , using, e.g., the description (6). (d) Express the coefficients of the inner products in i in terms of the metric coefficients g ij . (e) Use Tables A and B to express the smoothness conditions in terms of even functions φ i . (f) Row reduce the equations coming from all one-parameter groups L i , and replace some of the even functions in terms of others. (g) Solve the resulting system of equations for the metric g ij .

Example 1
A simple example is given by the groups G = Sp(1) × S 1 , K = {(e jθ , 1) | 0 ≤ θ ≤ 2π} · H and H Z 4 with generator (i, i). There exists an infinite family of inequivalent cohomogeneity one actions on S 5 as a special case of the Kervaire sphere examples, see [9], the simplest one being the tensor product action of SO(3)SO(2) on S 5 . For all of them one half of the group diagram is given by the above groups. Notice that the action of K on the slice V C is given by (q, z) · v = zv.
If we let X 1 = (i, 0), X 2 = (j, 0), X 3 = (k, 0) and Y = (0, i) then we have the Ad H invariant decomposition p = k = R · X 2 and m = m 0 ⊕ m 1 with m 0 = span{X 1 , Y }, m 1 = R · X 3 . Since Ad H acts as Id on m 0 and as − Id on p ⊕ m 1 the nonvanishing inner products are given by There is only the one-parameter group L = {exp(θX 2 ) | 0 ≤ θ ≤ 2π} to be considered. L acts via R(θ) on −1 = span{ċ(0), X 2 }, trivially on 0 = R · Y , and by R(2θ) on 1 = span{X 1 , X 3 }. Thus a = 1 and d 1 = 2. According to Tables B and C we have See also [8] Appendix 1 for a further class of examples with K/H S 1 .

Example 2
In [3], the author studied cohomogeneity one Ricci flat metrics on the homogeneous disk bundle with H = T 2 ⊂ K = U(2) ⊂ G = SU(3), where we assume that U(2) is the lower 2 × 2 block. We illustrate that the smoothness conditions can be obtained with our methods quickly.
Let E kl , iE kl , k < l, be the usual basis of su (3). Then the decomposition of h ⊥ into Ad H irreducible representations is given by: Since they are all inequivalent, the metric is determined by: The module p = n 1 is irreducible since K/H = S 2 and we can choose L = exp(θE 23 ). Since exp(πE 23 ) ∈ H, we have a = 2 and hence 1 2 E 23 has unit length in the Euclidean inner product g 0 on the slice. The decomposition under L is Since p is orthogonal to m, the decomposition of the slice V is not needed. Thus the metric is smooth if and only if for some smooth functions φ 1 , φ 2 .

Example 3
Let H ⊂ K ⊂ G be given by SO (2)  We consider the following basis of g = so (5): Then K 1 , K 2 , K 3 span the subalgebra k so(3) with [K 1 , K 2 ] = K 3 and cyclic permutations. Thus K i is orthonormal with respect to the biinvariant metric Q so(3) (A, B) = − tr(AB)/2 which induces the metric of constant curvature 1 on SO(3)/SO(2) = S 2 . We choose the base point such that the Lie algebra of its stabilizer group H is spanned by K 1 . Henceċ(0), K 2 , K 3 is an orthonormal basis in the inner product g 0 on V = R 3 . Notice that for the biinvariant metric Q so(5) (A, B) = − tr(AB)/2 we have Q so(5) (A, B) = 5Q so(3) (A, B) for A, B ∈ so(3). Thus, if we abbreviate Q = Q so(5) , we have Q(K i , K j ) = 5δ ij . On the other hand, V i are orthonormal unit vectors in Q.
We have the following decomposition of p ⊕ m as the sum of irreducible Hmodules: and all other scalar products are zero.
For the smoothness conditions, since Ad H acts irreducibly on p, we need to choose only one vector and set X = K 2 with L = exp(tK 2 ) ⊂ SO(3). Since SO(3) acts standard on V , we have a = 1. Furthermore, V = −1 ⊕ 0 with −1 = span{ċ(0), K 2 } and 0 = span{K 3 } since L acts via rotations in theċ(0), K 2 plane, and hence trivially on e 3 = K * 3 (0). Under the action of L, one easily sees that m decomposes as the sum of the following irreducible modules: and a Lie bracket computation shows that under the action of L we have d i = i for i = 1, 2, 3; e.g., . 1) Irreducible modules in m. We have three irreducible L-modules in m and for each of them we apply Lemma 5, and use the notation g ij therein. Notice that due to Ad H invariance, all vectors V i are orthogonal to each other. For 1 we have: Since d 1 = 1 and a = 1, we need For 2 we have: Since d 2 = 2, smoothness requires that For 3 we have: Since d 3 = 3, we need (15g 2 + g 4 ) + (10g 1 + 6g 3 ) = φ 5 (t 2 ), (15g 2 + g 4 ) − (10g 1 + 6g 3 ) = t 6 φ 6 (t 2 ).
In particular, all functions g 1 , g 2 , g 3 , g 4 are even, a fact that one could have already obtained from invariance of the metric under the Weyl group element.
2) Products between modules in m. Inner products between 0 and 2 , and between 1 and 3 are not necessarily 0. For the first one, Lemma 7 implies that √ and hence g 2 −g 4 = t 2 φ 7 (t 2 ), a condition already implied by K invariance at t = 0. For the second one, Lemma 6 and as well as 3) Smoothness on the slice. Section 3.1 implies that f = t 2 + t 4 φ(t 2 ) since a = 1. 4) Products between m and the slice V . All of the modules l i have nontrivial inner products with the slice. For the 4 inner products between −1 , i.e., K 2 , and i we get from Lemma 8: On the other hand, for the 4 inner products between 0 , i.e., K 3 , and i we get from Lemma 9: Thus we need:

5)
Combining all conditions. Summarizing the conditions in 1) and 2), we have for the inner products in m:

COHOMOGENEITY ONE MANIFOLDS
Notice that the last two conditions imply that (g 3 − g 1 ) = t 2 φ(t 2 ) and hence K invariance at t = 0 is encoded in the above equations. This is an over determined linear system of equations in the metric functions. Since we know there always exist solutions, we can row reduce in order to get the following relationships between the smooth functions: Thus necessary and sufficient conditions for smoothness in m are: which we can also solve for the metric and obtain (after renaming the even functions): for some smooth functions φ 1 , φ 2 , φ 3 , φ 4 of t 2 . Furthermore, f = t 2 + t 4 φ 5 (t 2 ), h 11 = t 4 φ 6 (t 2 ), h 12 = t 5 φ 7 (t 2 ).
We only need to consider the one-parameter group L = exp(tA) with A = 0 1 −1 0 . Since A is conjugate to diag(i, −i), the decomposition under L has the same weights and multiplicity. Thus in the description of the metric, we have exponents t k for k = 1, . . . , 4n − 2.

Proof of Theorem B
After we saw how the process works in concrete examples, we will now prove Theorem B. One needs to first derive all smoothness conditions obtained from Section 3, possibly for several circles L i . This gives rise to a highly over-determined system of equations for the r metric functions g ij , i ≤ j of the form for some smooth functions φ k . The coefficients a k ij do not depend on the metric, but only on the Lie groups involved. We first want to show that each metric function must be involved in at least one equation and hence N ≥ r. For this let w ∈ K be a Weyl group element. Recall that w is defined by w(ċ(0)) = −c(0) which defines it uniquely mod H. Furthermore, w normalizes H and w 2 ∈ H. Let n ⊂ p ⊕ m be an irreducible module under the action of H. Then we have either w(n) = n or w(n) = n with n another irreducible module invariant under H and equivalent to n.
If w(n) = n and X, Y ∈ n then Q(X, Y ) = Q(wX, wY ) and hence implies that g(X * , Y * ) is an even function.
We can now row reduce the systems, which we denote for short A k G = Φ. The last N − r rows in A k will consist of zeroes which implies that there exists a linear homogeneous relationship between the even functions φ k . Solving for one of the variables, and substituting into Φ we obtain a system of r equations in r unknowns. In the row-reduced system we cannot have a further row of zeroes in A k since otherwise we can express the metric in terms of r − 1 even functions, contradicting that the metric on the regular part consists of r arbitrary functions. Thus A k has maximal rank r and we can solve for g ij in terms of the remaining even functions. This proves Theorem B.

Actions on spheres
In order to facilitate the applications of determining the smoothness conditions in examples, we discuss here the choice for the vectors X, the decomposition of the action by L = exp(tX) on the slice, and the integers a, d i . Since L ⊂ K 0 , we can assume that K is connected. Although the action of K on V can be highly ineffective, there exists a normal subgroup containing L acting almost effectively and transitively on the sphere in V . In Table A we list the almost effective transitive actions by connected Lie groups on spheres. The effective actions and the decomposition of p into irreducibles one can, e.g., find in [18], and from this one easily derives the ineffective ones using representation theory.
Recall that the inclusion p ⊂ V is determined by the action fields of the action of K on V . For each irreducible module we choose a vector X ∈ p i and normalize X such that L = {exp(θv) | 0 ≤ θ ≤ 2π} ⊂ K is a closed one-parameter group. Furthermore, the integer a = |L ∩ H| is the ineffective kernel of the action of L on V and V is the sum of two-dimensional L invariant modules: . with a, d i ∈ Z, which we can assume to be positive.
We choose a basis e 1 , e 2 , · · · of V and the geodesics c(t) = te 1 . We now discuss each transitive action, one at a time, using the numbering in Table A. 1) K/H = SO(n + 1)/SO(n) = S n K acts by matrix multiplication x → Ax on V = R n+1 with orthonormal basis e 1 , e 2 , . . . , e n+1 . We choose the geodesic such that c(t) = te 1 and let H be the stabilizer group of e 1 , i.e., H = {diag(1, A) | A ∈ SO(n)}.
2 ) K/H = U(1)/Z k = S 2n+1 We list here separately the common case of K = U(1) acting on C as w → z k w with stabilizer group Z k the k-th roots of unity. Here p = p 0 spanned by F = i with F * = kie 1 . Thus −1 = {ċ(0), (1/k)F } with a = k.

COHOMOGENEITY ONE MANIFOLDS
The stabilizer group at i is given by the complex linear automorphisms, which is equal to SU(3). Thus its Lie algebra h is given by the constraints x i + x i+1 = y i + y i+1 = 0 for i = 1, 3, 5, and the complement p by Since the action of Ad H on p is irreducible, it is sufficient to consider only one one-parameter group, and we choose F = 2E 12 − E 47 + E 56 ∈ p with L = exp(θF ). It acts as a rotation in the e 4 , e 7 plane and e 5 , e 6 plane at speed 1, and in the e 1 , e 2 plane at speed 2, and as Id on e 3 . Thus The embedding Spin(7) ⊂ SO (8), and hence the action of K on the slice, is given by the spin representation. On the Lie algebra level we can describe this as follows. A basis of g 2 ⊂ so (8)  Since the action of Ad H on p is irreducible, we need to consider only one oneparameter group and we choose L = {exp(θF ) with F = E 12 + E 56 . It acts as a rotation in the e 1 , e 2 plane and e 5 , e 6 plane at speed 1, and as Id on e 3 , e 4 , e 7 , e 8 . Thus The embedding of H in K is given by the spin representation of Spin (7) in Spin(8) followed by the (lift of) the standard block embedding of Spin (8) in Spin (9). Let S ij be the standard basis of spin(9) under the isomorphism so (9) spin(9) and denote by E i,j the standard basis of so (16). Furthermore, Spin(9) acts on the slice V R 16 via the spin representation and one easily computes the image of S ij in so (16). We only need the basis of p = p 1 ⊕ p 2 .
For the smoothness conditions we need to choose two one-parameter groups. For L 1 = exp(θZ 2 ) we obtain