In the present chapter we discuss fundamental matters such as the Dirichlet energy and tension tensor of a unit tangent vector field
X on a Riemannian manifold (
M,
g), first and second variation formulae and the harmonic vector fields system, to be applied in the remaining chapters of this book. Indeed one may look at a smooth vector field
as a map of Riemannian manifolds (
M,
g) and (
T(
M),
G
s) where
G
s is the Sasaki metric (cf.
Chapter 1
) and apply the ordinary results in variational calculus to the functional
as a map of Riemannian manifolds (
M,
g) and (
T(
M),
G
s) where
G
s is the Sasaki metric (cf.
Chapter 1
) and apply the ordinary results in variational calculus to the functional
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