LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES

1. Introduction

Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural projection map

π : G → G/H

leads to a description of G as the frame bundle of G/H, and the set of frames over a given point x ∈ G/H is isomorphic to H. The fundamental property of homogenous spaces is that for any two points x, y ∈ G/H and any frames F_{x}, F_{y} based at x and y respectively, there is a symmetry of G/H that takes x to y and F_{x} to F_{y}.

However, there are many interesting spaces that are not homogenous. In this lecture we will consider the case of a Riemannian manifold M. In general M is not a homogenous space, but the method of moving frames can still be applied to the study of submanifolds of M.

2. Frames and connections on Riemannian manifolds

Let M be an n-dimensional Riemannian manifold. Then M is a differentiable manifold with a smoothly varying inner product ⟨, ⟩_{x} on each tangent space T_{x}M. An orthonormal frame at the point x ∈ M is an orthonormal basis {e_{1}, …, e_{n}} of the tangent space T_{x}M. The set of orthonormal frames at each point is isomorphic to the Lie group O(n), and the set of orthonormal frames on M forms a principal bundle over M with fiber O(n), called the frame bundle of M and denoted F(M). An orthonormal frame on the open set U ⊂ M is a choice of an orthonormal frame {e_{1}(x), …, e_{n}(x)} at each point x ∈ U such that each e_{i} is a smooth local section of the tangent bundle T M; any such frame is a smooth section of the frame bundle F(M).

What changes when M is not a homogenous space? First of all, while each fiber of the frame bundle is isomorphic to O(n) and so has a group structure, there is no group structure on the entire frame bundle F(M). Moreover, given an orthonormal frame {e_{1}(x), …, e_{n}(x)} on an open set U ⊂ M, there is no natural way of thinking of the frame vectors e_{i}(x) as functions from M to a fixed vector space. Each e_{i}(x) takes values in the vector space T_{x}M, and while this space is isomorphic to 𝔼^{n}, there is no canonical isomorphism T_{x}M ≅ 𝔼^{n} and hence no natural way of regarding the e_{i}(x) as 𝔼^{n}-valued functions. Instead we must regard them as sections of the vector bundle T M.

This raises the question of how to differentiate the e_{i}(x). The exterior derivative d is defined for functions and differential forms that take values in a fixed vector space, but not for sections of vector bundles. In order to differentiate sections of vector bundles we need the notion of a connection on the vector bundle T M.

Definition: Let Γ(T M) denote the space of smooth local sections of T M. An affine connection ∇ on T M is a map

∇ : Γ(T M) × Γ(T M) → Γ(T M),

with ∇(w, v) denoted ∇_{w}v, that satisfies

1. ∇_{fw_{1}+gw_{2}}v = f∇_{w_{1}}v + g∇_{w_{2}}v

2. ∇_{w}(v_{1} + v_{2}) = ∇_{w}v_{1} + ∇_{w}v_{2}

3. ∇_{w}(fv) = w(f)v + f∇_{w}v

for v, v_{i}, w, w_{i} ∈ Γ(T M) and f, g smooth real-valued functions on M.

By the linearity properties, a connection ∇ is completely determined by its action on any given frame {e_{1}, …, e_{n}} on M. Associated to ∇ and to the frame are scalar-valued 1-forms ω_{j}^{i}, 1 ≤ i, j ≤ n on M, called the connection forms, which are uniquely determined by the condition that for any w ∈ T_{x}M,

∇_{w}e_{i} = ∑_{j=1}^{n} e_{j} ω_{i}^{j}(w).

We can think of the connection ∇ as defining a map from vector fields v ∈ Γ(T M) to T M-valued 1-forms by letting ∇v be the 1-form defined by

(∇v)(w) = ∇_{w}v.

Then we have

∇e_{i} = ∑_{j=1}^{n} e_{j} ω_{i}^{j}

and by the linearity properties, if v = ∑ v^{i}e_{i}, then

∇v = ∑_{i=1}^{n}(e_{i} dv^{i} + v^{i} ∑_{j=1}^{n} e_{j} ω_{i}^{j})

= ∑_{i=1}^{n} e_{i}(dv^{i} + ∑_{j=1}^{n} v^{j}ω_{j}^{i}).

∇ is the analog of the exterior derivative d.

A connection ∇ is called symmetric if for any vector fields v, w ∈ Γ(T M),

∇_{v}w − ∇_{w}v = [v, w]

where [v, w] denotes the usual Lie bracket of vector fields. A connection ∇ is said to be compatible with the metric on M if for any vector fields v, w ∈ Γ(T M),

d⟨v, w⟩ = ⟨∇v, w⟩ + ⟨v, ∇w⟩.

Theorem: (Levi-Civita) Given a Riemannian manifold M, there exists a unique connection ∇ on T M which is both symmetric and compatible with the metric. This connection is called the Levi-Civita connection on T M.

From now on we will assume that ∇ is the Levi-Civita connection on T M.

Now suppose that {e_{1}, …, e_{n}} is an orthonormal frame on an open set in M, and let {ω_{j}^{i}, 1 ≤ i, j ≤ n} be the connection forms associated to the frame. Because the connection is compatible with the metric, we have

0 = d⟨e_{i}, e_{j}⟩

= ⟨∇e_{i}, e_{j}⟩ + ⟨e_{i}, ∇e_{j}⟩

= ⟨∑_{k=1}^{n} e_{k} ω_{i}^{k}, e_{j}⟩ + ⟨e_{i}, ∑_{k=1}^{n} e_{k} ω_{j}^{k}⟩

= ω_{i}^{j} + ω_{j}^{i}.

Therefore we have ω_{i}^{j} = −ω_{j}^{i}, just as in the homogenous case.

In addition to the connection forms ω_{j}^{i}, we also have the dual forms ω^{i} defined by the equation

dx = ∑_{i=1}^{n} e_{i} ω^{i}.

The forms {ω^{i}, ω_{j}^{i}} form a basis for the space of 1-forms on the frame bundle F(M). In order to compute the structure equations for these forms, we will first need to differentiate the equation

dx = ∑_{i=1}^{n} e_{i} ω^{i}. (2.1)

This requires some care. Since all the terms in these equations are 1-forms that take values in T M, we must use the connection ∇ to differentiate them. First consider the left-hand side of equation (2.1). In terms of any local coordinate system (x^{1}, …, x^{n}) on M we have

dx = ∑_{i=1}^{n} \frac{∂}{∂x^{i}} dx^{i}.

64 JEANNE NIELSEN CLELLAND

Because ∇ is symmetric and [\frac{∂}{∂x^{i}} , \frac{∂}{∂x^{j}}] = 0 we have

∇(dx) = ∑_{i=1}^{n} ∇(\frac{∂}{∂x^{i}}) ∧ dx^{i}

= ∑_{i=1}^{n} ∑_{j=1}^{n} ∇_{∂/∂x^{j}}(\frac{∂}{∂x^{i}}) dx^{j} ∧ dx^{i}

= ∑_{i<j}[∇_{∂/∂x^{i}}(\frac{∂}{∂x^{j}}) − ∇_{∂/∂x^{j}}(\frac{∂}{∂x^{i}})] dx^{i} ∧ dx^{j}

= 0.

Thus differentiating equation (2.1) yields

0 = ∑_{i=1}^{n}[∇e_{i} ∧ ω^{i} + e_{i} dω^{i}]

= ∑_{i=1}^{n}[∑_{j=1}^{n} e_{j} ω_{i}^{j} ∧ ω^{i} + e_{i} dω^{i}]

= ∑_{i=1}^{n} e_{i} (dω^{i} + ∑_{j=1}^{n} ω_{j}^{i} ∧ ω^{j}).

Therefore,

dω^{i} = − ∑_{j=1}^{n} ω_{j}^{i} ∧ ω^{j}. (2.2)

Next we want to compute the exterior derivatives of the ω_{j}^{i}. Differentiating equation (2.2) yields

0 = − ∑_{j=1}^{n}(dω_{j}^{i} ∧ ω^{j} + ∑_{k=1}^{n} ω_{j}^{i} ∧ ω_{k}^{j} ∧ ω^{k})

= − ∑_{j=1}^{n}(dω_{j}^{i} + ∑_{k=1}^{n} ω_{k}^{i} ∧ ω_{j}^{k}) ∧ ω^{j}. (2.3)

By a higher-degree version of Cartan’s lemma, there must exist 1-forms α_{jk}^{i} such that

dω_{j}^{i} + ∑_{k=1}^{n} ω_{k}^{i} ∧ ω_{j}^{k} = ∑_{k=1}^{n} α_{jk}^{i} ∧ ω^{k}.

Furthermore, I claim that we can choose the α_{jk}^{i} so that α_{jk}^{i} = −α_{ik}^{j}. It follows from the skew-symmetry of the ω_{j}^{i} that

∑_{k=1}^{n}(α_{jk}^{i} + α_{ik}^{j}) ∧ ω^{k} = 0.

By Cartan’s lemma, this implies that

α_{jk}^{i} + α_{ik}^{j} = ∑_{ℓ=1}^{n} T_{jkℓ}^{i}ω^{ℓ}

where T_{jkℓ}^{i} = T_{jℓk}^{i}. Moreover, T_{jkℓ}^{i} = T_{ikℓ}^{j}. Now the α_{jk}^{i}’s are only determined up to transformations of the form

α̃_{jk}^{i} = α_{jk}^{i} + ∑_{ℓ=1}^{n} U_{jkℓ}^{i}ω^{ℓ}

with U_{jkℓ}^{i} = U_{jℓk}^{i}. So by changing the α_{jk}^{i} if necessary, we can assume that T_{jkℓ}^{i} ≡ 0, and so α_{jk}^{i} = −α_{ik}^{j}.

Now since the forms {ω^{i}, ω_{j}^{i}} form a basis for the 1-forms on the frame bundle F(M), there must exist functions R_{jkl}^{i}, S_{jkm}^{iℓ} = −S_{jkℓ}^{im} such that

α_{jk}^{i} = − ∑_{ℓ=1}^{n} R_{jkl}^{i} ω^{ℓ} + ∑_{ℓ,m=1}^{n} S_{jkm}^{iℓ} ω_{ℓ}^{m}.

Substituting these equations into (2.3) yields

∑_{j,k,ℓ=1}^{n} R_{jkℓ}^{i} ω^{k} ∧ ω^{ℓ} ∧ ω^{j} + ∑_{j,k,ℓ,m=1}^{n} S_{jkm}^{iℓ} ω_{ℓ}^{m} ∧ ω^{k} ∧ ω^{j} = 0.

The vanishing of the terms involving ω_{ℓ}^{m} implies that S_{jkm}^{iℓ} = S_{kjm}^{iℓ}. But because S_{jkm}^{iℓ} is skew-symmetric in i and j, this implies that the S_{jkm}^{iℓ} must all be zero. Thus we have

dω_{j}^{i} = − ∑_{k=1}^{n} ω_{k}^{i} ∧ ω_{j}^{k} + ∑_{k,ℓ=1}^{n} R_{jkl}^{i}ω^{k} ∧ ω^{ℓ}. (2.4)

Without loss of generality we can assume that R_{jkℓ}^{i} = −R_{jℓk}^{i}; then the functions R_{jkℓ}^{i} are uniquely determined. They form the components of a tensor, called the Riemannian curvature tensor of the metric on M. In addition to the symmetries

R_{jkℓ}^{i} = −R_{ikℓ}^{j}, R_{jkℓ}^{i} = −R_{jℓk}^{i}

equation (2.3) implies that

R_{jkℓ}^{i} + R_{kℓj}^{i} + R_{ℓji}^{i} = 0.

This is called the Bianchi identity. Together with the other symmetries, it implies that

R_{jkℓ}^{i} = R_{ℓij}^{k}.

3. Surfaces in 3-dimensional Riemannian manifolds

Suppose that M is a 3-dimensional Riemannian manifold, and let x : Σ → M be an embedded surface in M. How much of the analysis from the case when M = 𝔼^{3} will remain valid in this more general setting?

We can still choose an orthonormal frame {e_{1}(x), e_{2}(x), e_{3}(x)} for T_{x}M at each point x ∈ Σ such that e_{3} is orthogonal to the tangent plane T_{x}Σ. For such a frame, e_{1} and e_{2} form a basis for the tangent plane T_{x}Σ, and the same reasoning as before implies that ω^{3} = 0 and that ω^{1}, ω^{2} form a basis for the 1-forms on Σ. The metric of M naturally induces a metric on Σ, and the first fundamental form of this metric is

I = (ω^{1})^{2} + (ω^{2})^{2}.

As in the Euclidean case, differentiating the equation ω^{3} = 0 and applying Cartan’s Lemma implies that there exist functions h_{11}, h_{12}, h_{22} on Σ such that

[\begin{matrix}ω_{1}^{3} \\ ω_{2}^{3} \end{matrix}] = [\begin{matrix}h_{11} & h_{12} \\ h_{12} & h_{22} \end{matrix}] [\begin{matrix}ω^{1} \\ ω^{2} \end{matrix}] .

The second fundamental form of Σ is given by

II = ω_{1}^{3} ω^{1} + ω_{2}^{3} ω^{2} = h_{11}(ω^{1})^{2} + 2h_{12}ω^{1}ω^{2} + h_{22}(ω^{2})^{2}.

As in the Euclidean case, the first and second fundamental forms are invariants of the surface, so if two surfaces have different first and second fundamental forms then they cannot be equivalent via a symmetry of M. However, in general there is no analog of Important Lemmas 1 and 2 that allows us to decide when we have a complete set of invariants for a surface or when two surfaces are equivalent via a symmetry of M. In particular, given arbitrary quadratic forms I and II on an abstract surface Σ, checking that the structure equations are satisfied is not sufficient to guarantee that there exists an immersion x : Σ → M whose first and second fundamental forms are I and II. The best we can do is to arbitrarily specify one quadratic form; for instance, we have the following isometric embedding theorem which is due independently to Cartan and Janet.

Theorem: Let M be a 3-dimensional real analytic Riemannian manifold, and let Σ be a surface with a prescribed real analytic metric I. Then locally there exists an embedding x : Σ → M for which the metric on Σ induced from M agrees with I.

Exercises

1. Let M be a Riemannian manifold, and let {e_{1}, …, e_{n}} be a frame on an open set in M with associated dual forms ω^{1}, …, ω^{n}. Show directly that there exist unique 1-forms ω_{j}^{i} = −ω_{i}^{j} such that

dω^{i} = − ∑_{j=1}^{n} ω_{j}^{i} ∧ ω^{j}.

(Hint: suppose that there are two such sets of 1-forms ω_{j}^{i}, ω̄_{j}^{i} and use Cartan’s lemma. Then use the skew-symmetry of the ω_{j}^{i}’s to show that in fact ω_{j}^{i} = ω̄_{j}^{i}.) These are the connection forms associated to the Levi-Civita connection on M, and this argument shows that the Levi-Civita connection is uniquely determined.

2. Let M be the standard 3-sphere S^{3}. From Lecture 2, Exercise 6 it follows that the structure equations of the dual and connection forms on S^{3} are

dω^{1} = −ω_{2}^{1} ∧ ω^{2} + ω_{1}^{3} ∧ ω^{3}
dω^{2} = ω_{2}^{1} ∧ ω^{2} + ω_{2}^{3} ∧ ω^{3}
dω^{3} = −ω_{1}^{3} ∧ ω^{1} − ω_{2}^{3} ∧ ω^{2}
dω_{2}^{1} = ω_{1}^{3} ∧ ω_{2}^{3} + ω^{1} ∧ ω^{2}
dω_{1}^{3} = ω_{2}^{3} ∧ ω_{2}^{1} + ω^{3} ∧ ω^{1}
dω_{2}^{3} = −ω_{1}^{3} ∧ ω_{2}^{3} + ω^{3} ∧ ω^{2}.

Let x : Σ → S^{3} be an embedding. The induced metric on Σ has Gauss curvature K given by the equation

dω_{2}^{1} = K ω^{1} ∧ ω^{2}.

Show that

K = h_{11}h_{22} − h_{12}^{2} + 1

where the h_{ij} are the coefficients of the second fundamental form. Why is this equation different than in the Euclidean case?