LECTURE 1. BACKGROUND FROM DIFFERENTIAL GEOMETRY '109certain compatiblity conditions, we call the splitting a connection. (Such a split-
ting "connects" the fibers over nearby points of X.) We impose the compatibility
condition as follows.Definition 1.3. A connection on Eis a map A: TE --7 7r* E, i.e. a 1-form on E
with values in 7r* E, such that:• A is a splitting of the exact sequence (1.1), i.e. Ao i = 1 : 7r E --7 7r E.
• A commutes with multiplication by scalars, i.e. if,\ E e and m>. : E --7 E
is multiplication by >., then m~A =,\·A.
It is sometimes useful to write a connection in terms of a local trivialization
~ : Elu --7 U x en. Let e, ... , ~n be the n corresponding e-valued functions on
Elu, i.e. ~(e) = (7r(e), (e(e), ... ,~n(e))). The connection 1-form A now takes
values in e n ; let A a be the coordinates, i.e. A= (A^1 , ... , An). Then we can write
(1.2)
Here Aab is a 1-form pulled back from U; we can think of A as an End(en)-valued
1-form on U. (In (1.2) and elsewhere, repeated indices are summed.)Exercise 1.4. Show that any connection A over U can be written in the form
(1.2), with Aab pulled back from U. Show conversely that for any Aab pulled back
from U, the 1-form A defined by (1.2) is a connection for Elu-Exercise 1.5. Show that the space of all connections on E is an affine space
over 01 (X,End(E)). In other words, if A is a connection, then a 1-form A' E
01 (E, 7r E) is also a connection if and only if there exists a E 01 (X, End(E)) such
that (A' - A)e = (7ra)(e).
A connection gives us a way to differentiate sections of E.Definition 1.6. Given a connection A, we define the covariant derivative
V' A : C^00 (E) --7 C^00 (T* x Q9 E)as follows. If 'ljJ E C^00 ( E), then V' A 'ljJ : T x X --7 E x is the composition
TxX ~ T,p(x)E ~ (7r* E),p(x) ____!!_______, E x.
Here is another way to understand this definition. The "horizontal" subspaceof TE is Ker(A). There is a map H: (7r*TX)e --7 TeE which sends a tangent vector
v E TX to its horizontal lift at e, namely the unique horizontal vector h E TeE
such that 7rh = v. Given a section 'ljJ: X --7 E and v E T xX, we can compare the
derivative of 'ljJ along v, namely 'ljJv, with the horizontal lift of v, namely Hv. The
difference 'ljJv - Hv is a vertical vector in 7r E, and this is the covariant derivative,
i.e.
V' A'l/J(v) = ir('ljJ*v - Hv).So the covariant derivative measures the deviation of a section from the horizontal
subspace.