Is f_m Smooth When f Is a Smooth Map Between Manifolds?

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SUMMARY

If f: M × N → P is a smooth map between smooth manifolds M, N, and P, then for a fixed m in M, the function f_m: N → P defined by f_m(n) = f(m, n) is indeed smooth. This conclusion aligns with the established fact that the smoothness of f implies the smoothness of the corresponding function f_m when M is a manifold. The discussion confirms that this property holds true universally for smooth maps between manifolds.

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Mathematicians, students of differential geometry, and anyone studying the properties of smooth maps between manifolds will benefit from this discussion.

slevvio
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Hello everyone, I just had a quick question I was hoping somebody could answer.

If f: M \times N \rightarrow P is a smooth map, where M,N and P are smooth manifolds, then is it true for fixed m that f_m : N \rightarrow P is smooth, where f_m (n) = f(m,n)?

Any help would be appreciated.
 
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Of course, since the equivalent statement when M=R^m, N=R^n is true.
 
ok thank you!
 

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