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Extra resources for A First Course in Differential Geometry

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As a rule in this book, as was stated in the introduction, interest is focused, at least initially, on properties of curves and surfaces in the small, and consequently a double point such as P in Fig. 1 is regarded as two distinct points, each of which lies on a different curve segment. 3. Change of Parameters It is often convenient to shift from one parameter representation of a curve to another. Quite generally, many important results in differential geometry can often be made direct and easy to achieve once a special parametric representation has been tactfully chosen.

II 16. The Jordan Theorem as a Problem in Differential Geometry in the Large The distinction between a problem in differential geometry in the large and a problem in the small for plane curves is that a finite extent of a curve is to be considered in the former case, and in addition some property of the curve is generally involved that cannot be studied without doing more than examining the curve in a neighborhood of each of its points. The distinction between the two kinds of problems is best brought out by giving concrete examples, as will be done in this and subsequent sections.

It is of interest to determine K when the curve is given by a parameter t other than «. For dijds # 0, the following formulas are obtained : ^ dt 'is' Since ds/dt = V(X')2 [cf. 18): so that , (X' x X*) x\x\ - 4x? , "(«h x u2) = ( X ,. X , ) V , = (x? )Vl (ux x ua), Sec. 12 EXISTENCE OF A PLANE CURVE FOR GIVEN CURVATURE K 27 This is the well-known formula for the curvature of a plane curve given in the parametric form xx = xx(t)9 x2 = x2(t). 20) it is readily checked that the curve X(t ) = (t2, t3) shown in Fig.

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