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By Werner Hildbert Greub, Stephen Halperin, James Van Stone

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Additional resources for Connections, curvature, and cohomology

Sample text

Yn Þ oF ðx0 Þ ¼ det ð2:16Þ Dðx1 ; . ; xn Þ oxj is the Jacobian. Employing higher derivatives, the Taylor expansion of a function F from the normed linear space X into a normed linear space Y reads 1 1 Fðx0 þxÞ ¼ Fðx0 ÞþDFðx0 Þxþ D2 Fðx0 ÞxxþÁÁÁþ Dk Fðx0 Þ xxÁÁÁx |fflffl{zfflffl} þÁÁÁ; ð2:17Þ 2! k! 3 Derivatives 25 provided x0 and x0 þ x belong to a convex domain X & X on which F is defined and has total derivatives to all orders, which are continuous functions of x0 in X and provided this Taylor series converges in the norm topology of Y.

Z. Then, DHðx0 Þ ¼ DGðFðx0 ÞÞ  DFðx0 Þ ð2:20Þ if the right hand side derivatives exist. In this case, DFðx0 Þ 2 LðX; YÞ and DGðFðx0 ÞÞ 2 LðY; ZÞ and hence DHðx0 Þ 2 LðX; ZÞ. Moreover, if DF : X ! LðX; YÞ is continuous at x0 2 X and DG : X0 ! LðY; ZÞ is continuous at Fðx0 Þ 2 X0 , then DH : X ! LðX; ZÞ is continuous at x0 2 X. Coming back to the warning on p. 23, take the function F : R ! R2 : t 7! ðt; t2 Þ; and for G : R2 ! R take the function of the example on p. 23. Then, HðtÞ ¼ ðG  FÞðtÞ ¼ t and hence DHð0Þ ¼ 1.

This restriction is a continuous function on X1 and hence is  0 since X1 is connected. Starting from every point of this subset, let now x2 run through X2 to obtain again F  0 for the restriction of F. After n steps, F  0 on X in contradiction to the assumption that F is surjective. h A concept seemingly related to connectedness but in fact independent is local connectedness. A topological space is called locally connected, if every point has a neighborhood base of connected neighborhoods. ) A connected space need not be locally connected.

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