Page 404 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 31: Joachimsthal's Notations

product associated with the first fundamental form, i.e., the Riemannian metric). In particular, Notes
v and w are constant and the angle between v(t) and w(t) is also constant.

The vector field v(t) is parallel if dv/dt is normal to the tangent plane to the surface X at (t),
and so

v'(t),w(t) = 0
for all t  I. Similarly, since w(t) is parallel, we have

v(t),w'(t) = 0

for all t  I. Then,

v(t),w(t) ' = v'(t),w(t)  v(t),w'(t)  0

for all t  I. which means that v(t),w(t) is constant along .

As a consequence of corollary 14.12.5, if  : I  X is a nonconstant geodesic on X, then   = c for
some constant c > 0.
Thus, we may reparametrize  w.r.t. the arc length s = ct, and we note that the parameter t of a
geodesic is proportional to the arc length of .

Lemma 8: Let  : I  X be a regular curve on a surface X, and for any t  I, let w  T (t ) X. Then,
0
0
there is a unique parallel vector field, w(t), along , so that w(t ) = w . 0
0
0
Lemma is an immediate consequence of standard results on ODE’s. This lemma yields the
notion of parallel transport.
Definition 7: Let  : I  X be a regular curve on a surface X, and for any t  I, let w  T (t 0 ) X. Let
0
0
w be the parallel vector field along , so that w(t ) = w , given by Lemma 8. Then, for any t  I,
0
0
the vector, w(t), is called the parallel transport of w along  at t.
0
It is easily checked that the parallel transport does not depend on the parametrization of . If X
is an open subset of the plane, then the parallel transport of w at t is indeed a vector w(t) parallel
0
to w (in fact, equal to w ).
0
0
However, on a curved surface, the parallel transport may be somewhat counter-intuitive.
If two surfaces X and Y are tangent along a curve,  : I  X, and if w  T (t ) X = T a(t ) Y is a tangent
0
0
0
vector to both X and Y at t , then the parallel transport of w along  is the same, whether it is
0
0
relative to X or relative to Y .
This is because Dw/dt is the same for both surfaces, and by uniqueness of the parallel transport,
the assertion follows.
This property can be used to figure out the parallel transport of a vector w when Y is locally
0
isometric to the plane.
In order to generalize the notion of covariant derivative, geodesic, and curvature, to manifolds
more general than surfaces, the notion of connection is needed.
If M is a manifold, we can consider the space, X(M), of smooth vector fields, X, on M. They are
smooth maps that assign to every point p  M some vector X(p) in the tangent space T M to M
p
at p.
We can also consider the set C (M) of smooth functions f : M   on M.
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