In geometric calculus, there is a concept of a vector manifold where the points are considered vectors in a general geometric algebra (a vector space with vector multiplication) which can then be shown to have the properties of a manifold (tangent spaces etc.). For a more precise definition of a vector manifold, see page 65 of
http://montgomerycollege.edu/Departments/planet/planet/Numerical_Relativity/bookGA.pdf
For what I believe to be the standard definition of a manifold see page 93 of the same book. This question doesn't seem to be answered inside the above book.
It seems that this approach is simpler than the traditional coordinate approach to manifolds since it is automatically independent of coordinates, except it might be less general. So, my question is: Are all manifolds to be vector manifolds or at least for every usual manifold to be isomorphic to some vector manifold? If not, are there simple additional criteria necessary to add to a manifold in order for it to be isomorphic to a vector manifold?
Note: I'm new to differential geometry, so feel free to correct anything above which may be incorrect. Also, thank you for any ideas/advice/answers you give.
Best Answer
The notes' definition of "vector manifold" in a geometric algebra $V$ is just as a subset $W\subset V$, but since he goes on to talk about smooth paths and tangent vectors, I expect that he means a smoothly embedded submanifold of $V$.
The Whitney embedding theorem guarantees that any smooth manifold can be smoothly embedded into Euclidean space of sufficiently high dimension. There are geometric algebras of arbitrarily high dimension (the Clifford algebra $Cl_n(\mathbb{R})$ is a vector space of dimension $2^n$). So any smooth manifold can be realized as a vector manifold.
The traditional abstract approach to defining manifolds is already automatically coordinate-free, and makes no reference to any ambient space. However, defining manifolds in ambient Euclidean space is equivalent to the coordinate-free approach - for example, standard textbooks such as Differential Topology (Guillemin-Pollack) define and work with manifolds entirely in ambient space.