Page 403 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Xuv =  1 12 X + G X + MN,
2
12
u
v
2
Xvu =  1 21 X + G X + MN,
u
21
v
2
Xvv =  1 22 X + G X + NN.
22
u
v
Now, Dw/dt is the tangential component of dw/dt, thus, by dropping the normal components,
we get
Dw 1 1 1 1 2 2 2 bu G bv)X
2
(a  






dt   11 au   12 av   21 bu   22 bv)X  (b   11 au   12 av   21   22  v
u
Thus, the covariant derivative only depends on y = (u , v), and the Christoffel symbols, but we
 
know that those only depends on the first fundamental form of X.
Definition 3: Let  : I  X be a regular curve on a surface X. A vector field along  is a map, w, that
assigns to every t  I a vector w(t)  T X in the tangent plane to X at (t). Such a vector field is
(t)
differentiable if the components a, b of w = aX + bX over the basis (X , X ) are differentiable. The
u
v
v
u
expression Dw/dt(t) defined in the above equation is called the covariant derivative of w at t.
Definition 4: extends immediately to piecewise regular curves on a surface.
Definition 5: Let  : I  X be a regular curve on a surface X. A vector field along  is parallel if
Dw/dt = 0 for all t  I.
Thus, a vector field along a curve on a surface is parallel if its derivative is normal to the surface.
For example, if C is a great circle on the sphere S parametrized by arc length, the vector field of
2
tangent vectors C’(s) along C is a parallel vector field.
We get the following alternate definition of a geodesic.
Definition 6: Let  : I  X be a nonconstant regular curve on a surface X. Then,  is a geodesic if
the field of its tangent vectors, (t), is parallel along , that is
 
D  0
dt (t) 
for all t  I.
If we let (t) = X(u(t), v(t)), from the equation

Dw 1 1 1 1  2 2 2 2 bv)X
(a  







u
dt   11 au   12 av   21 bu   22 bv)X  (b   11 au   12 av   21 bu   22  v
with a   and b   we get the equations
u
v,
2
2
1
u   1 11 (u)     1 21 uv   1 22 (v)  0



uv    
12
2
2
2

 v   2 11 (u)     2 21 uv   2 22 (v)  0,

uv    
12
1
1
2
which are indeed the equations of geodesics found earlier, since  =  and  =  2 21
12
21
12
(except that  is not necessarily parametrized by arc length).
Lemma 7: Let  : I  X be a regular curve on a surface X, and let v and w be two parallel vector
fields along . Then, the inner product v(t),w(t) is constant along  (where –, – is the inner
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