differential geometry w notes from teacher_Part_66

# differential geometry w notes from teacher_Part_66 - 131...

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5.4. DEGREE OF A MAP 131 It has a discrete non-negative real spectrum { λ } k = 0 bounded from below by zero 0 = λ 0 λ 1 λ 2 ··· with ﬁnite multiplicities. The eigenfunctions { h k } k = 0 form an orthonormal basis in L 2 ( V ), that is, ( h k , h l ) = Z V d vol h k h l = δ kl . For each f L 2 ( V ) there is a Fourier series f = X k = 0 a k h k , where a k = ( h k , f ) = Z V fh k . The lowest eigenvalue is 0. It is simple (has multiplicity 1). The corre- sponding eigenfunction is the constant h 0 = [vol ( V )] - 1 / 2 . Lemma 5.4.1 Let f be a function on a closed manifold M such that R M f = 0 . Let { λ k , h k } k = 1 be the spectral resolution of the operator ( - Δ ) . Then the equation Δ h = f has a unique solution given by the Fourier series h = - X k = 1 1 λ k a k h k , where a k = ( h k , f ) = Z M fh k . Proof : 1. ± di geom.tex; April 12, 2006; 17:59; p. 131

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132 CHAPTER 5. LIE DERIVATIVE Lemma 5.4.2 Let V be a closed (compact without boundary) oriented
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## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_66 - 131...

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