  We briefly discuss new models of an `affine' theory of gravity in
multidimensional space-times with symmetric connections. We use and generalize
Einstein's proposal to specify the space-time geometry by use of the Hamilton
principle to determine the connection coefficients from a geometric Lagrangian
that is an arbitrary function of the generalized Ricci curvature tensor and of
other fundamental tensors. Such a theory supplements the standard Einstein
gravity with dark energy (the cosmological constant, in the first
approximation), a neutral massive (or tachyonic) vector field vecton, and
massive (or tachyonic) scalar fields. These fields couple only to gravity and
can generate dark matter and/or inflation. The concrete choice of the geometric
Lagrangian determines further details of the theory. The most natural geometric
models look similar to recently proposed brane models of cosmology usually
derived from string theory.
