Page 402 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 31: Joachimsthal's Notations

If the geodesic polygon is a triangle, and if A,B,C are the interior angles, so that A =  –  , Notes
1
B =  – 2, C =  – 3, the Gauss-Bonnet theorem reduces to what is known as the Gauss formula:

 KdA  A B C   .


R
The above formula shows that if K > 0 on R, then  KdA is the excess of the sum of the angles
R
of the geodesic triangle over .

If K < 0 on R, then  KdA is the efficiency of the sum of the angles of the geodesic triangle
R
over .

And finally, if K = 0, then A + B + C = , which we know from the plane!
For the global version of the Gauss-Bonnet theorem, we need the topological notion of the
Euler-Poincare characteristic, but this is beyond the scope of this course.

31.2 Covariant Derivative, Parallel Transport, Geodesics Revisited

Another way to approach geodesics is in terms of covariant derivatives.
The notion of covariant derivative is a key concept of Riemannian geometry, and thus, it is
worth discussing anyway.

Let X :   E be a surface. Given any open subset, U, of X, a vector field on U is a function, w, that
3
assigns to every point, p  U, some tangent vector w(p)  T X to X at p.
p
A vector field, w, on U is differentiable at p if, when expressed as w = aX + bX in the basis
u
v
(X , X ) (of T X), the functions a and b are differentiable at p.
p
u
v
A vector field, w, is differentiable on U when it is differentiable at every point p  U.
Definition 3: Let, w, be a differentiable vector field on some open subset, U, of a surface X. For
every y  T X, consider a curve,  : ] – , [  U, on X, with (0) = p and a’(0) = y, and let
p
w(t) = (w  )(t) be the restriction of the vector field w to the curve . The normal projection of
dw/dt(0) onto the plane T X, denoted
p
Dw (0), or D ’0w(p), or D w(p),
dt  y
is called the covariant derivative of w at p relative to y.
The definition of Dw/dt(0) seems to depend on the curve , but in fact, it only depends on y and
the first fundamental form of X.

Indeed, if (t) = X(u(t), v(t)), from
w(t) = a(u(t), v(t))X + b(u(t), v(t))X ,
v
u
we get
dw = a(X u + X v) + b(X u + X v) + aXu + bXv.

dt uu  uv  vu  vv  
However, we obtained earlier the following formulae (due to Gauss) for X , X , X , and X :
uu
uv
vu
vv
2
X =  1 11 X + G X + LN,
u
11
v
uu
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