By Michael F. Atiyah (auth.)
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Extra info for Vector Fields on Manifolds
On the other hand, this simple situation is quite rare and it is rather a deep result that for spheres T Sn is equivalent to Sn × Rn only for n = 1, 3, 7 . For other spheres, their tangent bundles consist of twisted products of copies of Rn over Sn . In particular, T S2 is such a twisted product of S2 with one copy of R2 at each point. An intuitive picture of a 2-manifold that is a twisted obius strip, which we product of R1 (or an interval from it) over S1 is a M¨ know does not embed into R2 but does embed into R3 .
What we oﬀer here is some elementary geometry to display the features common, and of most signiﬁcance, to a wide range of typical statistical models for real processes. Many more geometrical tools are available to make further sophisticated studies, and we hope that these may attract the interest of those who model. For example, it would be interesting to explore the details of the role of curvature in a variety of applications, and to identify when the distinguished curves called geodesics, so important in fundamental physics, have particular signiﬁcance in various real K.
Next, we say that a map between manifolds f : M −→ N is diﬀerentiable at x ∈ M , if for some charts (U, φ) on M and (V, ψ) on N with x ∈ U, f (x) ∈ V , the map ψ ◦ f |U ◦ φ−1 : φ(U ) −→ ψ(V ) is diﬀerentiable as a map between subsets of Rn and Rm , if M is an n-manifold and N is an m-manifold. This property turns out to be independent of the choices of charts, so we get a linear map Tx f : Tx M −→ Tf (x) N . Moreover, if we make a choice of charts then Tx f appears in matrix form as the set of partial derivatives of ψ ◦ f ◦ φ−1 .
Vector Fields on Manifolds by Michael F. Atiyah (auth.)