Differential Geometry - Computer Graphics and Geometric Modeling - page 661

vector field along a curve that was defined in Section 9.10 and at the same time
specialize to that one.

Definition. A map

() ¥ () Æ ()

Vect MM M

XY

Vect Vect

( ) Æ—

, Y

X

is called a connection or covariant derivative on M if it satisfies the following proper-
ties. Let f Œ C • ( M ).

(1) — X Y is a bilinear function over R in X and Y .
(2) — f X Y = f — X Y .
(3) — X f Y = ( X f) Y + f — X Y , where X f( p ) is the directional derivative of f in the direc-
tion X at p .

The vector field — X Y is called the covariant derivative of the vector field Y with respect
to the vector field X .

Let

F : UM Æ

define a coordinate neighborhood for a point p Œ M . We shall use ∂/∂u i (same as F i )
to denote the standard basis vectors of the tangent spaces. Therefore in terms of this
basis,

n

Â

i k

—

∂∂ ∂∂

u = G ∂∂ u .

u j k i

k = 1

The functions G ij determine the connection completely and are called the Christoffel
symbols of the connection. In general, they are not the same functions that we defined
in Section 9.9 without some additional hypotheses. Given a connection on the mani-
fold M and a curve h(t) in M we can define a covariant derivative along curves.

Definition. Given a connection — X Y on M , a compatible covariant derivative along a
curve h: [a,b] Æ M is a map

D
dt

( ) Æ ( )

: Vect M , h Vect M , h

that sends a vector field X (t) along h into a new vector field (D X /dt)(t) along h satis-
fying the following properties. Assume that X (t), Y (t) Œ Vect( M ,h) and that f : [a,b] Æ
R is a differentiable function.

D
dt

D
dt

XY D

dt

( ) =+. (1) XY +

D
dt

XX X

df
dt

f D
dt

() =

(2) f + .