Global analysis. Exercises 10
1) Prove that if the connections with the Christoffel symbols k
ij and ~k
ij have equal
geodesics, then the connection with the Christoffel symbols k
ij + ~k
ij ( +  = 1)
has the same geodesics.
2) Solve the equation of the parallel displacement on the sphere with the metric
(d)2
+ sin2
(d)2
in the spherical coordinates:
* along a parallel ( = 0 = const);
* along a meridian ( = 0 = const).
3) Find the angle between a tangent vector to the sphere and its image under the
parallel displacement along the parallel.
4) Let M be a manifold with a torsion-free affine connection . Prove that if X
and Y are parallel vector fields (i.e. ZX = ZY = 0 for all vector fields Z), then
[X, Y ] = 0.
5) Let M be a manifold with a torsion-free affine connection . Prove that any
parallel distribution on M is involutive (a distribution is called parallel if Y X
belongs to this distribution for all X from this distribution and all vector fields Y
on M).
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