Lecture 24 - Curvature and Torsion on Principal Bundles (Schuller's Geometric Anatomy of Theoretical Physics)
Curvature and torsion on principal bundles
Usually, in more elementary treatments of differential geometry or general relativity, curvature and torsion are mentioned together as properties of a covariant derivative over the tangent or the frame bundle. Since we will soon define the notion of curvature on a general principal bundle equipped with a connection, one might expect that there be a general definition of torsion on a principal bundle with a connection. However, this is not the case. Torsion requires additional structure beyond that induced by a connection. The reason why curvature and torsion are sometimes presented together is that frame bundles are already equipped, in a canonical way, with the extra structure required to define torsion. 24.1
Covariant exterior derivative and curvature
Definition. Let (P, π, M ) be a principal G-bundle with connection one-form ω. Let φ be a k-form (i.e. an anti-symmetric, C ∞ (P )-multilinear map) with values in some module V . Then then exterior covariant derivative of φ is Dφ :
Γ(T P )×(k+1) → V (X1 , . . . , Xk+1 ) 7→ dφ(hor(X1 ), . . . , hor(Xk+1 )).
Definition. Let (P, π, M ) be a principal G-bundle with connection one-form ω. The curvature of the connection one-form ω is the Lie-algebra-valued 2-form on P Ω : Γ(T P ) × Γ(T P ) → Te G defined by Ω := Dω. For calculational purposes, we would like to make this definition a bit more explicit. Proposition 24.1. Let ω be a connection one-form and Ω its curvature. Then Ω = dω + ω
with the second term on the right hand side defined as (ω
ω)(X, Y ) := Jω(X), ω(Y )K
where X, Y ∈ Γ(T P ) and the double bracket denotes the Lie bracket on Te G. Remark 24.2. If G is a matrix Lie group, and hence Te G is an algebra of matrices of the same size as those of G, then we can write Ωij = dω ij + ω ik ∧ ω kj . Proof. Since Ω is C ∞ -bilinear, it suffices to consider the following three cases.
a) Suppose that X, Y ∈ Γ(T P ) are both vertical, that is, there exist A, B ∈ Te G such that X = X A and Y = X B . Then the left hand side of our equation reads Ω(X A , X B ) := Dω(X A , X B ) = dω(hor(X A ), hor(X B )) = dω(0, 0) = 0 while the right hand side is ω)(X A , X B ) = X A (ω(X B )) − X B (ω(X A )) q y − ω([X A , X B ]) + ω(X A ), ω(X B )
dω(X A , X B ) + (ω
= X A (B) − X B (A) − ω(X JA,BK ) + JA, BK
= − JA, BK + JA, BK
Note that we have used the fact that the map i : Te G → Γ(T P ) A 7→X A is a Lie algebra homomorphism, and hence X JA,BK = i(JA, BK) = [i(A), i(B)] = [X A , X B ], where the single square brackets denote the Lie bracket on Γ(T P ). b) Suppose that X, Y ∈ Γ(T P ) are both horizontal. Then we have Ω(X, Y ) := Dω(X, Y ) = dω(hor(X), hor(Y )) = dω(X, Y ) and (ω
ω)(X, Y ) = Jω(X), ω(Y )K = J0, 0K = 0.
Hence the equation holds in this case.
c) W.l.o.g suppose that X ∈ Γ(T P ) is horizontal while Y = X A ∈ Γ(T P ) is vertical. Then the left hand side is Ω(X, X A ) := Dω(X, X A ) = dω(hor(X), hor(X A )) = dω(hor(X), 0) = 0. while the right hand side gives dω(X, X A ) + (ω
ω)(X, X A ) = X(ω(X A )) − X A (ω(X)) q y − ω([X, X A ]) + ω(X), ω(X A ) = X(A) − X A (0) − ω(X JA,BK ) + J0, AK
= −ω([X, X A ]) = 0,
where the only non-trivial step, which is left as an exercise, is to show that if X is horizontal and Y is vertical, then [X, Y ] is again horizontal. We would now like to relate the curvature on a principal bundle to (local) objects on the base manifold, just like we have done for the connection one-form. Recall that a connection one-form on a principal G-bundle (P, π, M ) is a Te G-valued one-form ω on P . By using the notation Ω1 (P ) ⊗ Te G for the collection (in fact, bundle) of all Te G-valued one-forms, we have ω ∈ Ω1 (P ) ⊗ Te G. If σ ∈ Γ(T U ) is a local section on M , we defined the Yang-Mills field ω U ∈ Ω1 (U ) ⊗ Te G by pulling ω back along σ. Definition. Let (P, π, M ) be a principal G-bundle and let Ω be the curvature associated to a connection one-form on P . Let σ ∈ Γ(T U ) be a local section on M . Then, the two-form Riem ≡ F := σ ∗ Ω ∈ Ω2 (U ) ⊗ Te G is called the Yang-Mills field strength. Remark 24.3. Observe that the equation Ω = dω + ω σ ∗ Ω = σ ∗ (dω + ω ∗
σ ∗ ω.
= σ (dω) + σ (ω ∗
ω on P immediately gives
= d(σ ω) + σ ω Since Riem is a two-form, we can write Riemµν = (dω U )µν + ωµU
In the case of a matrix Lie group, by writing Γijµ := (ω U )ijµ , we can further express this in components as Riemijµν = ∂ν Γijµ − ∂µ Γijν + Γikµ Γkjν − Γikν Γkjµ from which we immediately observe that Riem is symmetric in the last two indices, i.e. Riemij[µν] = 0. Theorem 24.4 (First Bianchi identity). Let Ω be the curvature two-form associated to a connection one-form ω on a principal bundle. Then DΩ = 0. Remark 24.5. Note that since Ω = Dω, Bianchi’s identity can be rewritten as D2 Ω = 0. However, unlike the exterior derivative d, the covariant exterior derivative does not satisfy D2 = 0 in general.
Definition. Let (P, π, M ) be a principal G-bundle and let V be the representation space of a linear (dim M )-dimensional representation of the Lie group G. A solder(ing) form on P is a one-form θ ∈ Ω1 (P ) ⊗ V such that (i) ∀ X ∈ Γ(T P ) : θ(ver(X)) = 0; (ii) ∀ g ∈ G : g B ((C g)∗ θ) = θ; (iii) T M and PV are isomorphic as associated bundles. A solder form provides an identification of V with each tangent space of M . Example 24.6. Consider the frame bundle (LM, π, M ) and define θ : Γ(T (LM )) → Rdim M X 7→ (u−1 π(X) ◦ π∗ )(X) where for each e := (e1 , . . . , edim M ) ∈ LM , ue is defined as ∼
Rdim M − → Tπ(e) M
(x1 , . . . , xdim M ) 7→ xi ei . To describe the inverse map u−1 e explicitly, note that to every frame (e1 , . . . , edim M ) ∈ LM , there exists a co-frame (f 1 , . . . , f dim M ) ∈ L∗ M such that ∼
u−1 → Rdim M e : Tπ(e) M − Z 7→ (f 1 (Z), . . . , f dim M (Z)). Definition. Let (P, π, M ) be a principal G-bundle with connection one-form ω and let θ ∈ Ω1 (P ) ⊗ V be a solder form on P . Then Θ := Dθ ∈ Ω2 (P ) ⊗ V is the torsion of ω with respect to θ. Remark 24.7. You can now see that the “extra structure” required to define the torsion is a choice of solder form. The previous example shows that there a canonical choice of such a form on any frame bundle bundle. We would like to have a similar formula for Θ as we had for Ω. However, since Θ and θ are both V -valued but ω is Te G-valued, the term ω θ would be meaningless. What we have, instead, is the following Θ = dθ + ω θ, where the half-double wedge symbol intuitively indicates that we let ω act on θ. More precisely, in the case of a matrix Lie group, recalling that dim G = dim Te G = dim V , we have Θi = dθi + ω ik θk .
Theorem 24.8 (Second Bianchi identity). Let Θ be the torsion of a connection one-form ω with respect to a solder form θ on a principal bundle. Then DΘ = Ω
Remark 24.9. Like connection one-forms and curvatures two-forms, a torsion two-form Θ can also be pulled back to the base manifold along a local section σ as T := σ ∗ Θ. In fact, this is the torsion that one typically meets in general relativity.