By Michael Spivak

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**Extra info for A Comprehensive Introduction to Differential Geometry, VOL. 3, 2ND EDITION **

**Example text**

Sobolev Spaces 40 with 1/p = 1/q - 1/n and n q K(n, q) = qn - q n(q - 1)] 1 r(n + 1) 19 I 1 /n [r(n/q)I (n + 1 - n/qkv" _ 1 for 1

42). Given a tic y through Q orthogonal to (C) at Q, so that ] - e, + E[ a 2 y(A)EM, with y(0) = Q, set Yo = (dy/dA)x=o. 1) is equal to r, for all A. By (10), the second variation of d(P, )(2)) at A = 0 is 1(Y), where Y is the Jacobi field along (C) satisfying Y(P) = 0, Y(Q) = Yo. But [g(Y', Y') - b2g(Y, Y)] ds = Ib(Y); 1(Y) ? 0 4,(Y) is the index form (14) on a manifold with constant sectional curvature b2. On such a manifold, the solutions of (11) vanishing at s = 0 are of the type 4V = f sin bs, for i >: 2, where ff are some constants.

M As (V1V1 f + (l/n)Afg;1)(V1V f + (l/n)Afg'1) >- 0, it follows that V,V1 fV'Vif >- (1/n)(Af )2, hence 2(1 - 1/n) >- k. Chapter 2 Sobolev Spaces §1. 1 We are going to define Sobolev spaces of integer order on a Riemannian manifold. First we shall be concerned with density problems. Then we shall prove the Sobolev imbedding theorem and the Kondrakov theorem. After that we shall introduce the notion of best constant in the Sobolev imbedding theorem. , Hi on n-dimensional manifolds). For Sobolev spaces on the open sets in n-dimensional, real Euclidean space W", we recommend the very complete book of Adams [1].

### A Comprehensive Introduction to Differential Geometry, VOL. 3, 2ND EDITION by Michael Spivak

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