FINSLER GEOMETRY OVER THE REALS 11
(whenever
M
is) lets us make sense of certain averages that are built into these
equations.
Given the Ricci tensor, the criterion for generalized Einstein manifolds ([AZ2])
is then naturally a statement on
SM
which reads
Ricij
=
.gij·
Arguments involv-
ing Euler's theorem show that this equation holds if and only if ).. equals the Ricci
scalar and has no y-dependence (though it can depend on x). In other words, unlike
the Riemannian case, the ).. here is typically a function (rather than a constant)
on
M.
One can now ask, for example, whether in dimension
2:
3 every manifold
admits a Finsler
structure which satisfies
Ricij
=
.(x) 9ij,
or perhaps even one
with constant ).. ? (This is known to be false when posed for Riemannian metrics.)
[A1]
[A2]
[AIM]
[Ak]
[AI]
[AZ1]
[AZ2]
[BC1]
[BC2]
[BCS1]
[BCS2]
[Be]
References
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___ , Uber eine Verallgeminerung der Riemannschen Geometri, Forschunginst. Math.
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a
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=
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