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We construct a duality manifest gravitational theory for the special linear group, SL(N ) with N = 4. The spacetime is formally extended, to have the dimension 1 2 N (N − 1), yet is gauged. Consequently the theory is subject to a section condition. We introduce a semi-covariant derivative and a semi-covariant 'Riemann' curvature, both of which can be completely covariantized after symmetrizing or contracting the SL(N ) vector indices properly. Fully covariant scalar and 'Ricci' curvatures then constitute the action and the 'Einstein' equation of motion. For N ≥ 5, the section condition admits duality inequivalent two solutions, one (N − 1)-dimensional and the other three-dimensional. In each case, the theory can describe not only Riemannian but also non-Riemannian backgrounds.

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