Differentiable Manifolds
What’s the fuss?
In a simplistic senses spacetime can be encoded as a set, the elements of which correspond to spacetime points (Isham 1999).
Additional structure must be imposed however. A topological space encapsulates the basic notions of nearness, without quantifying what that nearness is.
But ultimately we want to be bring our calculus toolkit to this set which represents our spacetime.
Ultimately we have two aims:
- To be able to locate spacetime points via a set of real numbers.
- To be able to perform calculus related operations to study the dynamical evolution of systems.
From charts to atlases to differentiable manifolds
Remember that our calculus works on certain sets of real numbers called Euclidean Space. Hence we need mathematical machines that can lead us back and forth from the world of curved spaces to the flat Euclidean lands.
Let’s take a top down approach.
An atlas is a family of coordinate charts. A coordinate chart is pair (U,\phi)
A topological space is a differentiable manifold if it has a differential structure on it in the form of a complete atlas. Complete here simply means that the atlas is maximal i.e. not contained in any other atlas.
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