2. (a) If E1, E2 is a tangent frame field on M with connection form 012, show that = E2 [012(E)] - E (0,2(E)] -0,2(E.) -

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2 A If E1 E2 Is A Tangent Frame Field On M With Connection Form 012 Show That E2 012 E E 0 2 E 0 2 E 1 (19.38 KiB) Viewed 33 times

2 A If E1 E2 Is A Tangent Frame Field On M With Connection Form 012 Show That E2 012 E E 0 2 E 0 2 E 2 (3.6 KiB) Viewed 33 times

2. (a) If E1, E2 is a tangent frame field on M with connection form 012, show that = E2 [012(E)] - E (0,2(E)] -0,2(E.) -0,2(E2). (Hint: Write 02 = f10 + f20, where fi = 0;2(E;), and use Cor. 2.3.) (b) Check this formula on the sphere in Example 1.6. =

2.3 Corollary do 2 = -KO, A 0.. e

1.6 Example The sphere. Consider the adapted frame field E1, E2, E3 defined on the (doubly punctured) sphere & in Example 1.3. By extending this frame field to an open set of R' we get the spherical frame field given in Example 6.2 of Chapter 2, provided the indices of the latter are shifted by 1 → 3,2 – 1,3 → 2. Thus, in terms of the spherical coordinate functions, Example 8.4 of Chapter 2 gives 0, = r cos o du, 012 = sin o du, 02 = r do, 013 = -cos o du, 023 = -do. Because all forms (including functions) are now restricted to the surface , the spherical coordinate function p has become a constant: the radius r of the sphere. In general, the forms associated with an adapted frame field obey the fol- lowing remarkable set of equations.