  We classify the existent Birkhoff-type theorems into four classes: First, in
field theory, the theorem states the absence of helicity 0- and spin 0-parts of
the gravitational field. Second, in relativistic astrophysics, it is the
statement that the gravitational far-field of a spherically symmetric star
carries, apart from its mass, no information about the star; therefore, a
radially oscillating star has a static gravitational far-field. Third, in
mathematical physics, Birkhoff's theorem reads: up to singular exceptions of
measure zero, the spherically symmetric solutions of Einstein's vacuum field
equation with Lambda = 0 can be expressed by the Schwarzschild metric; for
Lambda unequal 0, it is the Schwarzschild-de Sitter metric instead. Fourth, in
differential geometry, any statement of the type: every member of a family of
pseudo-Riemannian space-times has more isometries than expected from the
original metric ansatz, carries the name Birkhoff-type theorem. Within the
fourth of these classes we present some new results with further values of
dimension and signature of the related spaces; including them are some
counterexamples: families of space-times where no Birkhoff-type theorem is
valid. These counterexamples further confirm the conjecture, that the
Birkhoff-type theorems have their origin in the property, that the two
eigenvalues of the Ricci tensor of two-dimensional pseudo-Riemannian spaces
always coincide, a property not having an analogy in higher dimensions. Hence,
Birkhoff-type theorems exist only for those physical situations which are
reducible to two dimensions.
