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A Special Stokes’s Theorem For Some Incomplete Riemannian Manifolds

Alavizadeh, Arian | 2016

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 48831 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Bahraini, Alireza
  7. Abstract:
  8. Let (M; g) be a Riemannian manifold. Using classical Stokes’ theorem one can show that the equality (dω; η)L2 = (ω; δη)L2 holds for smooth forms ! and η with compact supports, where δ is the formal adjoint of d . There are some examples of Riemannian manifolds for which the above equality does not hold for general forms ! and η i:e: smooth square-integrable forms such taht d! and δη are also squareintegrable. In the case that the above equality holds for such general forms on a Riemannian manifold (M; g) , we say that the L2 - Stokes theorem holds for (M; g) . In 1952, Gaffney showed that the L2 - Stokes theorem holds for complete Riemannian manifolds. But at that time, there was no powerful mathematical tool to study the L2 - Stokes theorem for incomplete Riemannian manifolds. At the late 70s, using purely functional analytic tools, Cheeger found a Necessary and sufcient condition for the L2 - Stokes theorem to be hold on a Riemannian manifold (M; g) . Cheeger also showed that for some special incomplete Riemannian manifolds known as manifolds with conic singularities, the L2 - Stokes theorem does hold. Using Cheeger’s approach, one can also study the Hodge theory and Poincaré duality for the case of L2 - Stokes theorem. Later, Lesch and Brüning as well as Hunsicker and Mazzeo generalized Cheeger’s results and proved that the L2 - Stokes theorem holds for Riemannian manifolds with simple edge singularities. In this thesis, we will have a review on the works by the people mentioned above
  9. Keywords:
  10. Conical Singularity ; Cohomology Group ; Riemannian Manifold ; Stokes Theorem ; Hilbert Complex ; Simple Edge Singularity

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