On Bent Manifolds and Deformed Spaces

Abstract

Riemannian geometry is the geometry of bent manifolds. However, as this paper shows in both a tangible and a formal way, it is also the geometry of deformed spaces. Applying Riemannian geometry to zones of our 3D space enables us to understand General Relativity (GR) almost intuitively, and inspires our imagination. Space in GR is considered a continuous manifold, bent (curved) by energy/momentum. Both Einstein (1933) [1] and Feynman (1963) [2], however, considered the option of space being a deformed (curved) continuum rather than a bent (curved) continuous manifold. In the GDM [3], though, space is a 3D deformed lattice rather than a bent continuous manifold.