Page 384 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

position of that point, allowing the ellipticity of the Earth to be calculated from measurements Notes
of gravity at different latitudes.

Formula

Clairaut’s formula for the acceleration of gravity g on the surface of a spheroid at latitude , was:

  5  2 
g  G 1   m f sin   ,



  2  
where G is the value of the acceleration of gravity at the equator, m the ratio of the centrifugal
force to gravity at the equator, and f the flattening of a meridian section of the earth, defined as:
a b

f  ,
a
(where a = semimajor axis, b = semiminor axis).
Clairaut derived the formula under the assumption that the body was composed of concentric
coaxial spheroidal layers of constant density. This work was subsequently pursued by Laplace,
who relaxed the initial assumption that surfaces of equal density were spheroids. Stokes showed
in 1849 that the theorem applied to any law of density so long as the external surface is a
spheroid of equilibrium.
The above expression for g has been supplanted by the Somigliana equation:

 1 k sin  
2

g  G , 
2
2
  1 e sin   

where, for the Earth, G = 9.7803267714 ms ; k =0.00193185138639; e = 0.00669437999013.
2
–2
30.6 Gauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important
statement about surfaces which connects their geometry (in the sense of curvature) to their
topology (in the sense of the Euler characteristic). It is named after Carl Friedrich Gauss who was
aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published
a special case in 1848.

30.6.1 Statement of the Theorem

Suppose M is a compact two-dimensional Riemannian manifold with boundary M. Let K be the
Gaussian curvature of M, and let k be the geodesic curvature of M. Then
g

 KdA  M  k ds  2 (M),

g
x
M 
where dA is the element of area of the surface, and ds is the line element along the boundary of
M. Here, (M) is the Euler characteristic of M.
If the boundary M is piecewise smooth, then we interpret the integral M  k ds as the sum of the
 g
corresponding integrals along the smooth portions of the boundary, plus the sum of the angles
by which the smooth portions turn at the corners of the boundary.

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