Geometry

Geometry of the Massless Sector

If the full closed-string massless sector is gravitational, then its geometry must accommodate \(g_{\mu\nu}\), \(B_{\mu\nu}\), and \(\phi\) without treating them as unrelated objects.

The proposed geometric data are the generalized metric and the DFT dilaton. They are intended to encode the closed-string massless sector in one geometric language.

The role of \(O(D,D)\) covariance is not decorative. It is the symmetry principle that makes the full sector geometric. This is the first test of Hypothesis B: the theory must organize the complete sector while preserving the Riemannian descriptions known to be correct in their domains.

In this program, doubled-yet-gauged spacetime is not a decorative enlargement of coordinates. It is the setting in which coordinate gauge symmetry, \(O(D,D)\) covariance, and the closed-string massless sector can be treated as parts of one geometric structure.

Connection and Curvature

A gravitational theory must define parallel transport. In Riemannian geometry this role is played by the Levi-Civita connection. In the closed-string setting, the corresponding connection must be compatible with the doubled geometric structure and with \(O(D,D)\) covariance.

Once such a connection exists, curvature becomes unavoidable. The resulting curvature must be geometric, covariant in the appropriate sense, and reducible to familiar Riemannian curvatures after a sector is chosen.

Covariant derivatives are therefore not optional notation. They are the mechanism by which fields, sources, and propagation are compared across the doubled-yet-gauged geometry.

Non-Riemannian Structures

The working hypothesis should not assume in advance that every admissible sector is Riemannian. Non-Riemannian geometry tests whether the DFT description is only a repackaging of familiar geometry or a broader framework for closed-string gravitational structure.

In this restrained sense, non-Riemannian structures are not added for novelty. They are part of the question of whether closed-string gravity has sectors that cannot be fully described by an ordinary Riemannian metric from the start.