Three-dimensional Manifolds as Slices of Infinite-dimensional Hilbert Manifolds: A Geometric Model of the Universe

Three-dimensional Manifolds as Slices of Infinite-dimensional Hilbert Manifolds

Abstract

This paper presents a geometric framework in which our observable three-dimensional universe is modeled as a smooth submanifold—specifically, a slice—embedded within an infinite-dimensional Hilbert manifold. Drawing on classical embedding theorems by Whitney, Nash, Kuiper, and Henderson, we reinterpret established results in differential topology through a novel lens that bridges geometry and physics. We demonstrate that any smooth 3-manifold can be realized as an isometric leaf in a smooth foliation of an infinite-dimensional manifold, and construct such foliations explicitly using smooth normal vector fields along a fixed embedding. We prove that the space of these foliations, parametrized by such fields, forms an infinite-dimensional Fréchet manifold—effectively a moduli space of parallel universes, each represented as a geometric slice. An explicit example using the 3-sphere SS3 embedded in the Hilbert space l 2 is developed, illustrating the theoretical construction in concrete terms. Diagrams and visualization accompany the model to clarify the geometric intuition and moduli variation. Our approach remains purely geometric, independent of physical field equations, yet conceptually resonates with brane world scenarios, emergent gravity, and infinite-dimensional quantum theories. This reinterpretation of classical geometry provides not only a rigorous mathematical result but also opens a pathway toward new models of dimensional emergence and foundational questions in cosmology and ontology. By positioning the universe as a geometric object embedded in an infinitedimensional ambient structure, we offer a new direction for thinking about space, structure, and reality.

Dhaka Univ. J. Sci. 73(2): 187-195, 2025 (July)