Page 413 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 31.6 Keywords

Exponential map: The exponential map can be used to define special local coordinate systems on
normal neighborhoods, by picking special coordinates systems on the tangent plane.
Polar coordinates can be used to prove the following lemma showing that geodesics locally
minimize arc length.
Gauss-Bonnet theorem: The local Gauss-Bonnet theorem deals with regions on a surface
homeomorphic to a closed disk, whose boundary is a closed piecewise regular curve  without
self-intersection.

31.7 Self Assessment


1. The principal normal n must be ................ to the normal N to the surface along the curve
segment from p to q.

2. Given a surface ................ a geodesic line, or geodesic, is a regular curve C : I  E on X,
3
such that k (t) = 0 for all t  I.
g
3. Given a surface X, for every point p = X(u, v) on X, for every non-null tangent vector

v T (u,v) (X) at p, there is some ................ and a unique curve  : ] –, [  E on the surface
3


X, such that  is a geodesic, (0) = p, and ’(0) = v.
4. In this case, ................ Thus, the closed half-line corresponding to  = 0 is omitted, and so
is its image under exp .
p
5. The local ................ deals with regions on a surface homeomorphic to a closed disk, whose
boundary is a closed piecewise regular curve  without self-intersection.

6. Let X :   E be a surface. Given any open subset, U, of X, a vector field on U is a function,
3
w, that assigns to every point, p  U, some tangent vector ................
31.8 Review Questions

1. Define Geodesic Lines, Local Gauss-Bonnet Theorem.
2. Discuss Covariant Derivative, Parallel Transport, Geodesics Revisited.
3. Describe Joachimsthal's Notations.

Answers: Self Assessment

1. parallel 2. X :   E ,
3
3.  > 0 4. 0 <  < 2.

5. Gauss-Bonnet theorem 6. w(p)  T X to X at p.
p

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