View Single Post
Feb5-12, 12:08 PM
P: 150
In my notes, the following two functions are defined:

Suppose [itex]M^m[/itex] and [itex]N^n[/itex] are smooth manifolds, [itex]F:M \to N[/itex] is smooth and [itex]p \in M[/itex]. We define:
[tex]F^*:C^\infty (F(p)) \to C^\infty (p)\ ,\ F^*(f) = f \circ F[/tex]
[tex]F_{*p}: T_pM \to T_{F(p)}N\ ,\ [F_{*p}(X)](f) = X(F^*f) = X(f \circ F)[/tex]

I understand the first function, [itex]F^*[/itex]; it maps [itex]f[/itex], a function on [itex]C^\infty(F(p))[/itex], to [itex]f \circ F[/itex], a function on [itex]C^\infty(p)[/itex].

However, I don't understand the second one, [itex]F_{*p}[/itex]. Since [itex]X(f) \in T_pM[/itex], it follows that [itex]f \in C^\infty (p)[/itex]. But then how is
[tex][F_{*p}(X)](f) = X(F^*f)[/tex]
defined? After all, in the definition of [itex]F_{*p}(X)[/itex], [itex]f[/itex] is a function on [itex] C^\infty (p)[/itex], not [itex]C^\infty(F(p))[/itex], so how can we evaluate [itex]F^*f[/itex]?
Phys.Org News Partner Science news on
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100