We briefly discuss new models of an `affine theory of gravity in multidimensional space-times with symmetric connections. We use and generalize Einsteins 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.
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