DRM: Directional Relational Manifolds

We introduce Directional Relational Manifolds (DRM), a class of geometrical structures inwhich dimensionality is not fixed but emerges dynamically from relational directional fields. Unlikeclassical manifolds based on orthogonal tangent spaces, DRMs define dimensions as active direc-tions whose structure may vary locally and globally. We develop the formal foundations of DRM,define its metric, connection, curvature, and dynamics, and show that stable DRMs naturallyconverge to toroidal topology. DRM extends information geometry Amari and Nagaoka(2000)from the fixed-dimension setting to variable-dimension structures, reducing to the Fisher–Raoframework as a special case. This framework provides a unified mathematical language foradaptive physical, cognitive, and informational systems.