Page 405 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Then, an affine connection, D, on M is a differentiable map,
D : X(M) × X(M)  X(M),

denoted DXY (or rXY ), satisfying the following properties:
(1) D fX+gY Z = fD Z + gD Z;
X
Y
(2) D (Y + Z) = D Y + D Z;
X
X
X
(3) D (fY ) = fD Y + X(f)Y ,
X
X
for all ,   , all X, Y, Z  X(M), and all f, g  C (M), where X(f) denotes the directional

derivative of f in the direction X.
Thus, an affine connection is C (M)-linear in X, -linear in Y, and satisfies a “Leibnitz” type of

law in Y.
For any chart  : U   , denoting the coordinate functions by x , . . . , x , if X is given locally by
m
1
m
m 
X(p) =  a (p) ,
i
i 1 x  i

then
m  (f   1 )

X(f)(p) =  a (p) .
i
i 1  x i

It can be checked that X(f) does not depend on the choice of chart.
The intuition behind a connection is that D Y is the directional derivative of Y in the direction X.
X
The notion of covariant derivative can be introduced via the following lemma:
Lemma 10: Let M be a smooth manifold and assume that D is an affine connection on M. Then,
there is a unique map, D, associating with every vector field V along a curve a : I  M on M
another vector field, DV/dt, along c, so that:

(1) D ( V   W)   DV   DW .

dt dt dt

(2) D (fV)  df V f DV .

dt dt dt
(3) If V is induced by a vector field Y  X(M), in the sense that V (t) = Y ((t)), then

DV D Y.
dt   '(t)
Then, in local coordinates, DV/dt can be expressed in terms of the Christoffel symbols, pretty
much as in the case of surfaces.

Parallel vector fields, parallel transport, geodesics, are defined as before.
Affine connections are uniquely induced by Riemmanian metrics, a fundamental result of Levi-
Civita.

In fact, such connections are compatible with the metric, which means that for any smooth curve
 on M and any two parallel vector fields X, Y along ”, the inner product X,Y i is constant.

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