**Authors:** Vu B Ho

In this work we discuss a geometric and topological quantisation of mass by extending our work on the principle of least action in which the quantisation of both the angular momentum of a Bohr hydrogen atom and the charge of an elementary particle can be shown to be quantised from the continuous deformation of a differentiable manifold. Similar to the case of the quantisation of charge using the two-dimensional Ricci scalar curvature, we show that in the case of three-dimensional differentiable manifolds we can apply the Yamabe problem, which states that any Riemannian metric on a compact smooth manifold of dimension greater or equal to three is conformal to a metric with constant scalar curvature, to show that mass can also be quantised by the deformation of differentiable manifolds. The Yamabe problem is a generalisation of the uniformisation theorem for two-dimensional differentiable manifolds. We also discuss whether quantum particles can be expressed as direct sums of quantum masses when they possess mathematical structures of differentiable manifolds, with the quantum masses are considered as prime manifolds. This may be regarded as a physical manifestation of an established mathematical proposition in differential geometry and topology that states that any compact, connected, and orientable differentiable manifold M can be decomposed into prime manifolds and the decomposition is unique up to an absorption or emission of 3-spheres S^3. This process of decomposition of differentiable manifolds into prime manifolds and the radiation of 3-spheres is similar to the radiation of the quanta of physical fields from a quantum system, such as the radiation of photons from a hydrogen atom. Even though our work is highly speculative and suggestive, we hope that it may pay way for further rigorous mathematical investigations into whether mass is also quantised like charge and other fundamental entities in physics.

**Comments:** 9 Pages.

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[v1] 2019-05-31 05:50:12

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