Geometric meaning of connections

Differential Geometry 2017–2018 ENS de Lyon

Geometric meaning of connections

Definition. A metric on a vector bundleE →M is a section h∈Γ(E^∗⊗E^∗) such that, for allx∈M,h_x is an inner product onE_x.

Definition. A connection∇ on a vector bundleE →M eqquiped with a metric h is said to becompatible withh if, for anys1, s2 ∈Γ(E),dh(s1, s2) =h(∇s₁, s2) +h(s1,∇s₂).

In the sequel, letp:E→M be a rankr vector bundle and let ndenote the dimension ofM. Definition. For any y∈E, we denoteVy = kerdyp. Then, V →E is a rank r sub-bundle of T E→E and is called thevertical sub-bundle ofT E.

Definition. An horizontal sub-bundle of T E is a sub-bundle H →E of T E →E such that, for anyy∈E,Hy⊗Vy =TyE.

Remarks. • Note thatH has rank nand that for anyy∈E,V_y =T_y(E_x)is canonically isomorphic to E_x, wherex=p(y).

• In the literature, horizontal sub-bundles are calledEhresmann connections, we don’t use this terminology in order to avoid confusing horizontal sub-bundles with connections (in the sense of the course).

For anyλ∈R, letM_λ :E →E denote the fiberwise multiplication by λ.

Let∆ :M →M×M be defined by ∆(x) = (x, x). Then∆^∗(E×E)→M is the space (x, y, y⁰)∈M×E×E

p(y) =x=p(y⁰) '

(y, y⁰)∈E×E

p(y) =p(y⁰)

with the natural projection. We denote byA: ∆^∗(E×E)→E the fiberwise addition ofE.

Definition. We say that an horizontal sub-bundle H of T E islinear if:

• for any y, y⁰∈E such thatp(y) =p(y⁰) we have:

d_(y,y⁰₎A (Hy×Hy⁰)∩T_(y,y⁰₎∆^∗(E×E)

=H_A(y,y⁰₎;

• for any λ∈Randy ∈E we haved_yM_λ(H_y) =H_Mλ(y).

The main goal of the following exercise is to prove that the choice of a linear horizontal sub-bundle ofT E is equivalent to the choice of a connection onE.

Exercise 1. Letp:E→M be a vector bundle of rank r on an-dimensional basis.

1. Let H → E be a linear horizontal sub-bundle of T E. Define a connection ∇ on E associated with H.

Hint: consider the projection ontoV_y alongH_y.

2. Conversely, assume thatE is eqquiped with a connection ∇. Define a linear horizontal sub-bundle H of T E associated with∇.

Hint: consider the image ofd_xs, wheres∈Γ(E) is such that ∇_xs= 0.

3. Recall that ifs(x) = 0 then∇_xsdoes not depend on∇. What does it mean in terms of the associated linear horizontal sub-bundles?

4. Lethbe a metric on E and let∇be a connection on E compatible withh. What does it mean for the associated linear horizontal sub-bundle?

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