Principal Bundles and Curvature Part Ii; Principal Bundles

PRINCIPAL BUNDLES AND CURVATURE PART II; PRINCIPAL BUNDLES 1. INTRODUCTION The1 plan for this section is as follows. First we will present principal bundles through the example of frame bundles of vector bundles. This has the advantage of a canonical system of coordinates. Then we will look at the more general form of the theory. There are several good reasons for moving from vector bundles to principal bundles. 1. Change of the local basis of sections is a group action on the vector bundle but in the vector bundle situation it is difficult to see what the group action is really doing. In principal bundles this knowledge becomes more systematic and we understand it better. 2. A connection on a vector bundle is associated with a particular local basis of sections. We have the formulas for changing to a new connection when we change to a new local basis of sections, but we understand this only in the most shallow way. In the Principle Bundle context we see that this is an action of the group on the Lie Algebra. 3. Once a connection is put on a principal bundle it metastasizes through all the associated bundles in a natural way. This replaces what look like haphazard and ad hoc methods for constructing connections on bundles related to a vector bundle. 4. The role of ”preserving a structure” in relation to connections is clearly reflected in the group of the principal bundle. Various properties like skew symmetry of the connection for the tangent bundle using an orthonormal basis in Riemannian Geometry become understandable as properties of the Lie Algebra of the group O(n, R). 5. It is a convenient mathematical setting in which to study Gauge Theory. These are fairly compelling reasons for introducing principal bundles. For amusement I will offer an analogy. The role of principal bundles in differential geometry might be compared to the role of evolution in biology. It is perfectly possible to work in a specific area of biology, for example turtles, without evolution. But the big scale structure of turtle theory will never be clear without evolution. Each new turtle requires a specific investigation. For a specific example in Differential Geometry, consider the following. It is natural to require in the Riemannian case that the covariant derivative satisfy d(σ, ρ) = (Dσ,ρ) + (σ,Dρ) 111 April 2012 1 If we come at this from the principal bundle approach, we don’t have to worry about whether it’s natural; it will automatically fall out of the theory when the group is O(n, R), which is the natural choice of group in the Riemannian context. 2. REVIEW OF VECTOR BUNDLES In this section we will briefly review what we know about Vector Bundles and digest the formulas for future use. All these were derived in Part I. Let π : E → M be a vector bundle. We will denote by σ = {σ1,...,σn} a local basis of sections over U ⊆ M and by {σ˜1,..., σñ} second such. We will denote individual sections over U by τ,ρ. Thus σα,τ,ρ ∈ Γ(U, E). Let {σα} be a local basis of sections over U. For any x ∈ U there is a vector −1 space π [x] isomorphic to E above x in E, and σ(x) = (σ1(x),...,σn(x)) will be a basis of this vector space. (The basis may be arbitrary or it may be a special kind of basis, for example orthonormal, depending on additional structures on the manifold, for example a Riemannian structure.) In general, if {σ˜α} is a second such local basis of sections, then they will be connected by a group α R element gβ (x) ∈ GL(n, ) so that α σ˜β = σαgβ (x) ∞ α We assume that everything is smooth (C ) so that we may regard (gβ (x)) as a local section of GL(n, R) over U. (Here we are thinking of GL(n, R) as being the trivial bundle U × GL(n, R) over U.) Now we wish to digest the material on Vector Bundles that we derived in Part I so it will be handy for reference. Vector Bundle (E,M,π) where π : E → M Fibre is E U coordinate patch of M {σ1,...,σn} fixed local basis of sections over U u1,...,ud local coordinates on U ρ is a local section of E and thus α 1 d ρ = σαρ (u ,...,u ) ρ has local coordinates u1,...,ud,ρ1,...,ρn (bundle coordinates) ∂ ∂ ∂ ∂ T (E) has a basis ,..., , ,..., ρ ∂u1 ∂ud ∂ρ1 ∂ρn ∂ ∂ v ∈ T (E) has a representation v = vi + V α ρ ∂ui ∂ρα ∂ w is in the VERTICAL space ⇔ w = W α ⇔ all wi =0 ∂ρα 2 The connection is characterized by its HORIZONTAL SPACE. We can define this by a projection Π at each point in E which projects the tangent space at that point of the bundle to the vertical space which is the tangent space of the fibre and a subset of the full tangent space. Given a connection, the projection Π is defined by i ∂ α ∂ α α β i ∂ Π v i + V α = V +Γβiρ v α ∂u ∂ρ ∂ρ Note how Π is a projection onto the Vertical Space. The Horizontal Space is defined as the kernel of Π; v is in the HORIZONTAL space ⇔ 0 = Πv ⇔ v1 1 . W Γ1 ρβ, ··· Γ1 ρβ, 1, 0, ··· 0 . 2 β 1 βd W Γ2 ρβ, ··· Γ2 ρβ, 0, 1, ··· 0 vd 0= . = β,1 βd . ····················· V 1 . n Γn ρβ, ··· Γn ρβ, 0, 0, ··· 1 . W β,1 βd . V n α β i β ⇔ 0=Γβiρ v + V A vector field X on M has the form ∂ X = Xi ∂ui The LIFT of X to HE is ∂ ∂ X˜ = Xi + X˜ α ∂ui ∂ρα where X˜ is in the Horizontal Space and thus 0 = ΠX˜ ˜ α α β i 0 = X +Γβiρ X so ∂ X˜ = Xi − Γα ρβXi ∂ui βi We wish to lift the curve x(t) from M to E. We do this by lifting it’s tangent vectorx ˙(t) to HE⊆TE by using the lift equations and then integrating along the lift to get the curve in E. (Technically, it would be best to embedx ˙(t) in a vector field in a neighborhood of the curve x(t), lift the whole field, and then pursue the integral curve in E.) Let’s look at these equations explicitly. Curve x(t) with coordinates ui(t) and 3 dui ∂ tangent vector X(t)= dt ∂ui dui ∂ dui ∂ Lift: X˜(t)= − Γα ρβ dt ∂ui βi dt ∂ρα Equations of the liftx ˜(t) of x(t) which is the integral curve of X˜(t) and has coordinatesu ˜i(t),ρα(t): du˜i dui = dt dt dρα dui = X˜ α = −Γα ρβ dt βi dt whereu ˜i(0) = ui(0) are the coordinates of x(0) so that πx˜(t)= x(t) and where ρα(0) are the coordinates of an arbitrary vector lying over x(0) in E. To further understand what the lift of the curve x(t) means, recall that α Dρ = σαDρ α α β = σα(dρ + ωβ ρ ) α α β i = σα(dρ +Γβiρ du ) Thus for a tangent vector v in T (E) α α β i j ∂ β ∂ Dρ(v) = σα(dρ +Γβiρ du ) v j + V β ∂u ∂ρ α α β i = σα V +Γβiρ v Since the tangent vector to the lift is dui ∂ dui ∂ x˜˙ = − Γα ρβ dt ∂ui βi dt ∂ρα we see Dρ(x˜˙ )=0 Just to be completely clear here, ρ is an element of the vector bundle E. It is determined by coordinates ui which determine the point of U over which ρ α i α α lies, and by the coordinates ρ . The u determine Γβi , and the Γβi and the ρα determine Dρ which is closely related to the projection Π from the Tangent Space of E to the Vertical Space of E. Note that ρ lives in E, not in the tangent space T (E). Now let’s review the equations for parallel transport of a vector ρ along a curve x(t) in U. We recall from part I that the equations for parallel transport α 0 = Dρ(x ˙) = σαDρ (x ˙) j α α β du ∂ = σα(dρ + ωβ ρ ) j dt ∂u 4 α j ∂ρ i α β i du ∂ = σα( i du +Γβiρ du ) j ∂u dt ∂u α i i ∂ρ du α β du = σα i +Γβiρ ∂u dt dt α i dρ α β du = σα +Γβiρ dt dt Thus we see that lifting x(t) tox ˜(t) is just another way of describing parallel transport of vectors. Finally, we recall that ∂ [X, Y ] − [X,˜ Y˜ ] = Ω α(X, Y )ρβ β ∂ρα ∂ = R α XiY j ρβ β ij ∂ρα Thus if one knows the horizontal space, one knows the lifts of vector fields and thus one knows the curvature. 3. FROM VECTOR BUNDLES TO FRAME BUNDLES Now we are ready to introduce the Principal Bundle. In the present context we have two pictures of the principle bundle, the first of which is called the frame bundle. 1. In the first picture we get the principal bundle by replacing the fibre π−1[x] of E by a fibre consisting of all the different bases of the fibre E of E. An element of the new fibre over x will be denoted by σ(x)= {σ1(x),...,σn(x)}.

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