3.2. The Energy of Unit Killing Fields in Dimension 3
Let
M
be a compact orientable
n
-dimensional Riemannian manifold. The greatest lower bound of
is
and this is achieved solely by the parallel unit tangent vector fields (when these exist). The problem of minimizing
E
is therefore more interesting when
M
admits no parallel unit vector fields, as for instance when
M
=
S
2
m
+1
or, more generally, when
M
is a compact Riemannian manifold of nonzero constant sectional curvature.
is
and this is achieved solely by the parallel unit tangent vector fields (when these exist). The problem of minimizing
E
is therefore more interesting when
M
admits no parallel unit vector fields, as for instance when
M
=
S
2
m
+1
or, more generally, when
M
is a compact Riemannian manifold of nonzero constant sectional curvature.
Our previous
Theorem 3.9
shows that unit Killing fields on a space ...
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