{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

differential geometry w notes from teacher_Part_98

differential geometry w notes from teacher_Part_98 -...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern