¯

Te (G/K). For k ∈ K we have p —¦ conjk = p —¦ »k —¦ ρk’1 = »k —¦ p and consequently

¯

¯

Te p—¦AdG,K (k) = Te p—¦Te (conjk ) = Te »k —¦Te p. Thus the isomorphism Te p : g/k ’

¯

Te (G/K) is K-equivariant for the representations Ad⊥ and Te » : k ’ Te »k .

¯ ¯

¯ ¯ ¯

¯

Now we consider the associated vector bundle G[Te (G/K), Te »] = (G —K

¯ ¯

Te (G/K), p, G/K, Te (G/K)), which is isomorphic to G[g/k, Ad⊥ ], since the rep-

¯ ¯

¯

resentation spaces are isomorphic. The mapping T2 » : G — Te (G/K) ’ T (G/K)

¯

(where T2 is the second partial tangent functor) is K-invariant and therefore

induces a mapping ψ as in the following diagram:

ee

q

G — Te (G/K)

¯

eg

e

¯

£

T»

ew T (G/K)

T (G/K)

ψ

(b) G —K

e

e

¯

ee

he π

e

p G/K

G/K.

This mapping ψ is an isomorphism of vector bundles.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

10. Principal ¬ber bundles and G-bundles 97

It remains to show the last assertion. The short exact sequence (a) induces a

sequence of vector bundles over G/K:

G/K — 0 ’ G[k, AdK ] ’ G[g, AdG,K ] ’ G[g/k, Ad⊥ ] ’ G/K — 0

This sequence splits ¬berwise thus also locally over G/K, so G[g/k, Ad⊥ ] •

G[k, AdK ] ∼ G[g, AdG,K ] and it remains to show that G[g, AdG,K ] is a trivial vec-

=

tor bundle. Let • : G—g ’ G—g be given by •(g, X) = (g, AdG (g)X). Then for

k ∈ K we have •((g, X).k) = •(gk, AdG,K (k ’1 )X) = (gk, AdG (g.k.k ’1 )X) =

(gk, AdG (g)X). So • is K-equivariant from the ˜joint™ K-action to the ˜on the

left™ K-action and therefore induces a mapping • as in the diagram:

¯

w G—g

•

G—g

u u

q

w G/K — g

ee •

¯

pr

(c) G —K g

ege £

p 1

G/K

The map • is a vector bundle isomorphism.

¯

10.16. Tangent bundles of Grassmann manifolds. From 10.5 we know

that (V (k, n) = O(n)/O(n ’ k), p, G(k, n), O(k)) is a principal ¬ber bundle.

Using the standard representation of O(k) we consider the associated vector

bundle (Ek := V (k, n)[Rk ], p, G(k, n)). It is called the universal vector bundle

over G(k, n). Recall from 10.5 the description of V (k, n) as the space of all linear

isometries Rk ’ Rn ; we get from it the evaluation mapping ev : V (k, n) — Rk ’

Rn . The mapping (p, ev) in the diagram

99

V (k, n) — Rk

99ev)

A

9

(p,

u

q

(a)

w G(k, n) — R

V (k, n) —O(k) Rk n

ψ

is O(k)-invariant for the action R and factors therefore to an embedding of

vector bundles ψ : Ek ’ G(k, n) — Rn . So the ¬ber (Ek )W over the k-plane W

in Rn is just the linear subspace W . Note ¬nally that the ¬ber wise orthogonal

complement Ek ⊥ of Ek in the trivial vector bundle G(k, n)—Rn with its standard

Riemannian metric is isomorphic to the universal vector bundle En’k over G(n’

k, n), where the isomorphism covers the di¬eomorphism G(k, n) ’ G(n ’ k, n)

given also by the orthogonal complement mapping.

Corollary. The tangent bundle of the Grassmann manifold is

T G(k, n) ∼ L(Ek , Ek ⊥ ).

=

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

98 Chapter III. Bundles and connections

Proof. We have G(k, n) = O(n)/(O(k) — O(n ’ k)), so by theorem 10.15 we get

— (so(n)/(so(k) — so(n ’ k))).

T G(k, n) = O(n)

O(k)—O(n’k)

On the other hand we have V (k, n) = O(n)/O(n ’ k) and the right action of

O(k) commutes with the right action of O(n ’ k) on O(n), therefore

V (k, n)[Rk ] = (O(n)/O(n ’ k)) — Rk = O(n) Rk ,

—

O(k) O(k)—O(n’k)

where O(n ’ k) acts trivially on Rk . Finally

L(Ek , Ek ⊥ ) = L O(n) Rk , O(n) Rn’k

— —

O(k)—O(n’k) O(k)—O(n’k)

L(Rk , Rn’k ),

—

= O(n)

O(k)—O(n’k)

where the left action of O(k) — O(n ’ k) on L(Rk , Rn’k ) is given by (A, B)(C) =

B.C.A’1 . Finally we have an O(k) — O(n ’ k) - equivariant linear isomorphism

L(Rk , Rn’k ) ’ so(n)/(so(k) — so(n ’ k)), as follows:

so(n)/(so(k) — so(n ’ k)) =

skew 0 A

A ∈ L(Rk , Rn’k )

= :

’At 0

skew 0

0 skew

10.17. The tangent group of a Lie group. Let G be a Lie group with

Lie algebra g. We will use the notation from 4.1. First note that T G is

also a Lie group with multiplication T µ and inversion T ν, given by (see 4.2)

T(a,b) µ.(ξa , ·b ) = Ta (ρb ).ξa + Tb (»a ).·b and Ta ν.ξa = ’Te (»a’1 ).Ta (ρa’1 ).ξa .

Lemma. Via the isomomorphism T ρ : g — G ’ T G, T ρ.(X, g) = Te (ρg ).X, the

group structure on T G looks as follows: (X, a).(Y, b) = (X + Ad(a)Y, a.b) and

(X, a)’1 = (’ Ad(a’1 )X, a’1 ). So T G is isomorphic to the semidirect product

g G, see 5.16.

Proof. T(a,b) µ.(T ρa .X, T ρb .Y ) = T ρb .T ρa .X + T »a .T ρb .Y =

= T ρab .X + T ρb .T ρa .T ρa’1 .T »a .Y = T ρab (X + Ad(a)Y ).

Ta ν.T ρa .X = ’T ρa’1 .T »a’1 .T ρa .X = ’T ρa’1 . Ad(a’1 )X.

Remark. In the left trivialisation T » : G — g ’ T G, T ».(g, X) = Te (»g ).X,

the semidirect product structure is: (a, X).(b, Y ) = (ab, Ad(b’1 )X + Y ) and

(a, X)’1 = (a’1 , ’ Ad(a)X).

Lemma 10.17 is a special case of 37.16 and also 38.10 below.

10.18. Tangent bundles and vertical bundles. Let (E, p, M, S) be a ¬ber

bundle. The subbundle V E = { ξ ∈ T E : T p.ξ = 0 } of T E is called the

vertical bundle and is denoted by (V E, πE , E).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

11. Principal and induced connections 99

Theorem. Let (P, p, M, G) be a principal ¬ber bundle with principal right ac-

tion r : P — G ’ P . Let : G — S ’ S be a left action. Then the following

assertions hold:

(1) (T P, T p, T M, T G) is again a principal ¬ber bundle with principal right

action T r : T P — T G ’ T P .

(2) The vertical bundle (V P, π, P, g) of the principal bundle is trivial as a

vector bundle over P : V P ∼ P — g.

=

(3) The vertical bundle of the principal bundle as bundle over M is again a

principal bundle: (V P, p —¦ π, M, T G).

(4) The tangent bundle of the associated bundle P [S, ] is given by

T (P [S, ]) = T P [T S, T ].

(5) The vertical bundle of the associated bundle P [S, ] is given by

V (P [S, ]) = P [T S, T2 ] = P —G T S, where T2 is the second partial

tangent functor.

Proof. Let (U± , •± : P |U± ’ U± — G) be a principal ¬ber bundle atlas with

cocycle of transition functions (•±β : U±β ’ G). Since T is a functor which

respects products, (T U± , T •± : T P |T U± ’ T U± — T G) is again a principal

¬ber bundle atlas with cocycle of transition functions (T •±β : T U±β ’ T G),

describing the principal ¬ber bundle (T P, T p, T M, T G). The assertion about

the principal action is obvious. So (1) follows. For completeness sake we include

here the transition formula for this atlas in the right trivialization of T G:

T (•± —¦ •’1 )(ξx , Te (ρg ).X) = (ξx , Te (ρ•±β (x).g ).(δ•±β (ξx ) + Ad(•±β (x))X)),

β

where δ•±β ∈ „¦1 (U±β ; g) is the right logarithmic derivative of •±β , see 4.26.

(2) The mapping (u, X) ’ Te (ru ).X = T(u,e) r.(0u , X) is a vector bundle iso-

morphism P — g ’ V P over P .

(3) Obviously T r : T P — T G ’ T P is a free right action which acts transitive on

the ¬bers of T p : T P ’ T M . Since V P = (T p)’1 (0M ), the bundle V P ’ M is

isomorphic to T P |0M and T r restricts to a free right action, which is transitive

on the ¬bers, so by lemma 10.3 the result follows.

(4) The transition functions of the ¬ber bundle P [S, ] are given by the expression

—¦(•±β —IdS ) : U±β —S ’ G—S ’ S. Then the transition functions of T (P [S, ])

are T ( —¦ (•±β — IdS )) = T —¦ (T •±β — IdT S ) : T U±β — T S ’ T G — T S ’ T S,

from which the result follows.

(5) Vertical vectors in T (P [S, ]) have local representations (0x , ·s ) ∈ T U±β —T S.

Under the transition functions of T (P [S, ]) they transform as T ( —¦ (•±β —

IdS )).(0x , ·s ) = T .(0•±β (x) , ·s ) = T ( •±β (x) ).·s = T2 .(•±β (x), ·s ) and this

implies the result.

11. Principal and induced connections

11.1. Principal connections. Let (P, p, M, G) be a principal ¬ber bundle.

Recall from 9.3 that a (general) connection on P is a ¬ber projection ¦ : T P ’

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

100 Chapter III. Bundles and connections

V P , viewed as a 1-form in „¦1 (P ; T P ). Such a connection ¦ is called a principal

connection if it is G-equivariant for the principal right action r : P — G ’ P , so

that T (rg ).¦ = ¦.T (rg ), i.e. ¦ is rg -related to itself, or (rg )— ¦ = ¦ in the sense

of 8.16, for all g ∈ G. By theorem 8.15.7 the curvature R = 1 .[¦, ¦] is then also

2

rg -related to itself for all g ∈ G.

Recall from 10.18.2 that the vertical bundle of P is trivialized as a vector

bundle over P by the principal action. So we have ω(Xu ) := Te (ru )’1 .¦(Xu ) ∈ g

and in this way we get a g-valued 1-form ω ∈ „¦1 (P ; g), which is called the

(Lie algebra valued) connection form of the connection ¦. Recall from 5.13 the

fundamental vector ¬eld mapping ζ : g ’ X(P ) for the principal right action.

The de¬ning equation for ω can be written also as ¦(Xu ) = ζω(Xu ) (u).

Lemma. If ¦ ∈ „¦1 (P ; V P ) is a principal connection on the principal ¬ber

bundle (P, p, M, G) then the connection form has the following properties:

(1) ω reproduces the generators of fundamental vector ¬elds, so we have

ω(ζX (u)) = X for all X ∈ g.

(2) ω is G-equivariant, so ((rg )— ω)(Xu ) = ω(Tu (rg ).Xu ) = Ad(g ’1 ).ω(Xu )

for all g ∈ G and Xu ∈ Tu P .

(3) For the Lie derivative we have LζX ω = ’ ad(X).ω.

Conversely a 1-form ω ∈ „¦1 (P, g) satisfying (1) de¬nes a connection ¦ on P

by ¦(Xu ) = Te (ru ).ω(Xu ), which is a principal connection if and only if (2) is

satis¬ed.

Proof. (1) Te (ru ).ω(ζX (u)) = ¦(ζX (u)) = ζX (u) = Te (ru ).X. Since Te (ru ) :

g ’ Vu P is an isomorphism, the result follows.

(2) Both directions follow from

Te (rug ).ω(Tu (rg ).Xu ) = ζω(Tu (rg ).Xu ) (ug) = ¦(Tu (rg ).Xu )

Te (rug ). Ad(g ’1 ).ω(Xu ) = ζAd(g’1 ).ω(Xu ) (ug) =

= Tu (rg ).ζω(Xu ) (u) = Tu (rg ).¦(Xu ).

(3) is a consequence of (2).

11.2. Curvature. Let ¦ be a principal connection on the principal ¬ber bundle

(P, p, M, G) with connection form ω ∈ „¦1 (P ; g). We already noted in 11.1 that

the curvature R = 1 [¦, ¦] is then also G-equivariant, (rg )— R = R for all g ∈ G.

2

Since R has vertical values we may again de¬ne a g-valued 2-form „¦ ∈ „¦2 (P ; g)

by „¦(Xu , Yu ) := ’Te (ru )’1 .R(Xu , Yu ), which is called the (Lie algebra-valued)

curvature form of the connection. We also have R(Xu , Yu ) = ’ζ„¦(Xu ,Yu ) (u). We

take the negative sign here to get the usual curvature form as in [Kobayashi-

Nomizu I, 63].

We equip the space „¦(P ; g) of all g-valued forms on P in a canonical way

with the structure of a graded Lie algebra by

[Ψ, ˜]§ (X1 , . . . , Xp+q ) =

1

= signσ [Ψ(Xσ1 , . . . , Xσp ), ˜(Xσ(p+1) , . . . , Xσ(p+q) )]g

p! q! σ

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

11. Principal and induced connections 101

or equivalently by [ψ — X, θ — Y ]§ := ψ § θ — [X, Y ]g . From the latter description

it is clear that d[Ψ, ˜]§ = [dΨ, ˜]§ + (’1)deg Ψ [Ψ, d˜]§ . In particular for ω ∈

„¦1 (P ; g) we have [ω, ω]§ (X, Y ) = 2[ω(X), ω(Y )]g .

Theorem. The curvature form „¦ of a principal connection with connection

form ω has the following properties:

(1) „¦ is horizontal, i.e. it kills vertical vectors.

(2) „¦ is G-equivariant in the following sense: (rg )— „¦ = Ad(g ’1 ).„¦. Conse-

quently LζX „¦ = ’ ad(X).„¦.

(3) The Maurer-Cartan formula holds: „¦ = dω + 1 [ω, ω]§ .

2

Proof. (1) is true for R by 9.4. For (2) we compute as follows:

Te (rug ).((rg )— „¦)(Xu , Yu ) = Te (rug ).„¦(Tu (rg ).Xu , Tu (rg ).Yu ) =

= ’Rug (Tu (rg ).Xu , Tu (rg ).Yu ) = ’Tu (rg ).((rg )— R)(Xu , Yu ) =

= ’Tu (rg ).R(Xu , Yu ) = Tu (rg ).ζ„¦(Xu ,Yu ) (u) =

= ζAd(g’1 ).„¦(Xu ,Yu ) (ug) =

= Te (rug ). Ad(g ’1 ).„¦(Xu , Yu ), by 5.13.

(3) For X ∈ g we have iζX R = 0 by (1), and using 11.1.(3) we get

1 1 1

iζX (dω + [ω, ω]§ ) = iζX dω + [iζX ω, ω]§ ’ [ω, iζX ω]§ =

2 2 2

= LζX ω + [X, ω]§ = ’ad(X)ω + ad(X)ω = 0.

So the formula holds for vertical vectors, and for horizontal vector ¬elds X, Y ∈

C ∞ (H(P )) we have

R(X, Y ) = ¦[X ’ ¦X, Y ’ ¦Y ] = ¦[X, Y ] = ζω([X,Y ])

1

(dω + [ω, ω])(X, Y ) = Xω(Y ) ’ Y ω(X) ’ ω([X, Y ]) = ’ω([X, Y ]).

2

11.3. Lemma. Any principal ¬ber bundle (P, p, M, G) admits principal con-

nections.

Proof. Let (U± , •± : P |U± ’ U± — G)± be a principal ¬ber bundle atlas. Let

us de¬ne γ± (T •’1 (ξx , Te »g .X)) := X for ξx ∈ Tx U± and X ∈ g. An easy

±

computation involving lemma 5.13 shows that γ± ∈ „¦1 (P |U± ; g) satis¬es the

requirements of lemma 11.1 and thus is a principal connection on P |U± . Now

let (f± ) be a smooth partition of unity on M which is subordinated to the open

cover (U± ), and let ω := ± (f± —¦ p)γ± . Since both requirements of lemma 11.1

are invariant under convex linear combinations, ω is a principal connection on

P.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

102 Chapter III. Bundles and connections

11.4. Local descriptions of principal connections. We consider a principal

¬ber bundle (P, p, M, G) with some principal ¬ber bundle atlas (U± , •± : P |U± ’

U± — G) and corresponding cocycle (•±β : U±β ’ G) of transition functions.

We consider the sections s± ∈ C ∞ (P |U± ) which are given by •± (s± (x)) = (x, e)

and satisfy s± .•±β = sβ .

(1) Let ˜ ∈ „¦1 (G, g) be the left logarithmic derivative of the identity,

i.e. ˜(·g ) := Tg (»g’1 ).·g . We will use the forms ˜±β := •±β — ˜ ∈

„¦1 (U±β ; g).

Let ¦ = ζ —¦ ω ∈ „¦1 (P ; V P ) be a principal connection with connection form

ω ∈ „¦1 (P ; g). We may associate the following local data to the connection:

(2) ω± := s± — ω ∈ „¦1 (U± ; g), the physicists version of the connection.

(3) The Christo¬el forms “± ∈ „¦1 (U± ; X(G)) from 9.7, which are given by

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g ).

(4) γ± := (•’1 )— ω ∈ „¦1 (U± — G; g), the local expressions of ω.

±

Lemma. These local data have the following properties and are related by the

following formulas.

(5) The forms ω± ∈ „¦1 (U± ; g) satisfy the transition formulas

ω± = Ad(•’1 )ωβ + ˜β± ,

β±

and any set of forms like that with this transition behavior determines a

unique principal connection.

(6) We have γ± (ξx , T »g .X) = γ± (ξx , 0g ) + X = Ad(g ’1 )ω± (ξx ) + X.

(7) We have “± (ξx , g) = ’Te (»g ).γ± (ξx , 0g ) = ’Te (»g ). Ad(g ’1 )ω± (ξx ) =

’T (ρg )ω± (ξx ), so “± (ξx ) = ’Rω± (ξx ) , a right invariant vector ¬eld.

Proof. From the de¬nition of the Christo¬el forms we have

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g )

= ’T (•± ).Te (r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’Te (•± —¦ r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’(0x , Te (»g )ω.T (•± )’1 (ξx , 0g )) = ’(0x , Te (»g )γ± (ξx , 0g )).

This is the ¬rst part of (7). The second part follows from (6).

γ± (ξx , T »g .X) = γ± (ξx , 0g ) + γ± (0x , T »g .X)

= γ± (ξx , 0g ) + ω(T (•± )’1 (0x , T »g .X))

= γ± (ξx , 0g ) + ω(ζX (•’1 (x, g))) = γ± (ξx , 0g ) + X.

±

So the ¬rst part of (6) holds. The second part is seen from

γ± (ξx , 0g ) = γ± (ξx , Te (ρg )0e ) = (ω —¦ T (•± )’1 —¦ T (IdX —ρg ))(ξx , 0e ) =

= (ω —¦ T (rg —¦ •’1 ))(ξx , 0e ) = Ad(g ’1 )ω(T (•’1 )(ξx , 0e ))

± ±

= Ad(g ’1 )(s± — ω)(ξx ) = Ad(g ’1 )ω± (ξx ).

Via (7) the transition formulas for the ω± are easily seen to be equivalent to the

transition formulas for the Christo¬el forms in lemma 9.7.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

11. Principal and induced connections 103

11.5. The covariant derivative. Let (P, p, M, G) be a principal ¬ber bundle

with principal connection ¦ = ζ —¦ ω. We consider the horizontal projection

χ = IdT P ’¦ : T P ’ HP , cf. 9.3, which satis¬es χ —¦ χ = χ, im χ = HP ,

ker χ = V P , and χ —¦ T (rg ) = T (rg ) —¦ χ for all g ∈ G.

If W is a ¬nite dimensional vector space, we consider the mapping χ— :

„¦(P ; W ) ’ „¦(P ; W ) which is given by

(χ— •)u (X1 , . . . , Xk ) = •u (χ(X1 ), . . . , χ(Xk )).

The mapping χ— is a projection onto the subspace of horizontal di¬erential forms,