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24

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

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.

1

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

= 0.

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,

2

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 ∗

ω) ∗

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

ωνU .

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.

3

24.2

Torsion

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

ue :

(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 .

4

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

θ.

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.

5

View more...
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

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.

1

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

= 0.

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,

2

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 ∗

ω) ∗

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

ωνU .

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.

3

24.2

Torsion

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

ue :

(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 .

4

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

θ.

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.

5

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