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Boman's theorem

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Differential geometry

differential geometry

synthetic differential geometry

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Boman's Theorem

Idea

In considering certain types of generalized smooth spaces, one may try to describe the smooth structure on a space by specifying the smooth curves. Boman's Theorem shows that this is sufficient to describe the smooth structure on a smooth manifold.

Statement

The following theorem appears (as Theorem 1) in the paper Boman 1967, where it is first proved, by Jan Boman:

Theorem

Let ff be a function from d\mathbb{R}^d to \mathbb{R}, and assume that the composed function fuf \circ u belongs to C (,)C^\infty(\mathbb{R},\mathbb{R}) for every uC (, d)u \in C^\infty(\mathbb{R}, \mathbb{R}^d). Then fC ( d,)f \in C^\infty(\mathbb{R}^d, \mathbb{R}).

Here, d\mathbb{R}^d is a Cartesian space, and C (X,Y)C^\infty(X,Y) is the set of smooth maps from XX to YY.

The theorem is quoted with a proof in Kriegl & Michor 1997 (Theorem 3.4).

Less than smooth functions

This theorem is for smooth functions, that is C C^\infty maps. A similar theorem could be stated for continuous functions, that is C 0C^0 maps. The situation is slightly less than ideal, however, for continuously differentiable functions, that is C 1C^1 maps, or more generally C pC^p maps for 0<p<0 \lt p \lt \infty.

Boman 1967 has this as part of Theorem 2:

Theorem

Let ff be a function from d\mathbb{R}^d to \mathbb{R}, and assume that the composed function fuf \circ u belongs to C p(,)C^p(\mathbb{R},\mathbb{R}) for every uC (, d)u \in C^\infty(\mathbb{R}, \mathbb{R}^d). Then fC p1( d,)f \in C^{p-1}(\mathbb{R}^d, \mathbb{R}).

Note that pp has become p1p - 1 in the conclusion. (Boman's full Theorem 2 gives stronger results involving Lipschitz conditions.)

Boman's Theorem 8 gives the desired result if we use parametrized surfaces instead of curves:

Theorem

Let ff be a function from d\mathbb{R}^d to \mathbb{R}, and assume that the composed function fuf \circ u belongs to C p( 2,)C^p(\mathbb{R}^2,\mathbb{R}) for every uC ( 2, d)u \in C^\infty(\mathbb{R}^2, \mathbb{R}^d). Then fC p( d,)f \in C^p(\mathbb{R}^d, \mathbb{R}).

Here we have 2\mathbb{R}^2 instead of \mathbb{R} as the domain of uu.

Boman's Theorem 3 guarantees such counterexamples as

f:x,yy 3x 2+y 2 f\colon x, y \mapsto \frac{y^3}{x^2 + y^2}

(continuously extended so that f(0,0)=0f(0,0) = 0). Given any smooth —or even C 1C^1— curve u:t(g(t),h(t))u\colon t \mapsto (g(t), h(t)), it may be shown (by several tedious cases) that (fu)(f \circ u)' is continuous. Nevertheless, ff is not C 1C^1 at (0,0)(0,0). (The general pattern, expressed in Boman's Theorem 10, is to use a function homogeneous in degree pp and C pC^p except at 0\vec{0} other than a polynomial. So long as d>1d \gt 1, such functions exist.)

This does not contradict the well known theorem (often taken as a definition!) that a function is C 1C^1 already if only its partial derivatives are continuous; while the partial derivatives of ff may be expressed as derivatives of fuf \circ u for appropriate smooth uu, the continuity of the partial derivatives requires not only that (fu)(t)(f \circ u)'(t) be continuous in tt but also that it be continuous in uu (and in fact jointly in tt and uu).

References

  • Jan Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 1967 249–268, MR237728 pdf

  • Andreas Kriegl, Peter W. Michor, The convenient setting of global analysis, Math. Surveys and Monographs 53, Amer. Math. Soc. 1997. x+618 pp. ISBN: 0-8218-0780-3 html MR1471480

Revised on February 7, 2014 22:59:54 by Toby Bartels (75.88.85.132)