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differential geometry w notes from teacher_Part_98

# differential geometry w notes from teacher_Part_98 -...

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7.5. HARMONIC FORMS 195 Corollary 7.5.1 Let M be a closed Riemannian manifold. Then any closed p-form β is a sum of an exact form d α and a harmonic form h, that is, β = d α + h . Proof : 1. Corollary 7.5.2 Let M be a closed Riemannian manifold. Then each de Rham class of cohomologous closed p-forms has a harmonic repre- sentative. Let k = B p ( M ) be the p-th Betti number. Let z (1) p , . . . , z ( k ) p , be a basis of real p-cycles in the real homology groups H p ( M ) and π 1 , . . . , π k be ar- bitrary real numbers. Then there is a unique harmonic p-form h p such that h p , z ( i ) p = π i , i = 1 , 2 , . . . , k . Proof : 1. The metric g is said to have positive Ricci curvature if its Ricci tensor is positive-definite. Corollary 7.5.3 Bochner Theorem Let M be a closed Riemannian manifold with positive Ricci curvature. Then the first Betti number van- ishes, i.e. B 1 ( M ) = 0 . That is there are no harmonic

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Unformatted text preview: 1-forms on M. Proof : 1. Let h be a harmonic 1-form. Then = 1 2 Z M Δ h h , h i vol = Z M R i j h i h j vol + ||∇ j h i || 2 ≥ . di ﬀ geom.tex; April 12, 2006; 17:59; p. 193 196 CHAPTER 7. HOMOLOGY THEORY 2. Thus h = 0. ± • Remark . The elements of the ﬁrst homology group H 1 ( M , G ) are equiva-lence classes of 1-cycles. • The 1-cycles are closed oriented curves (loops) on M . • If a closed curve can be deformed to a point, then it is a boundary of a surface (a 2-simplex). • That is, a closed curve that can be contracted to a point is a trivial 1-cycle. • • Corollary 7.5.4 For any simply connected manifold M and any Abelian group G, H 1 ( M , G ) = . di ﬀ geom.tex; April 12, 2006; 17:59; p. 194...
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