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

=

.gijĀ·

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]

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