*Charles P. Boyer and Krzysztof Galicki*

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0010
- Subject:
- Mathematics, Geometry / Topology

This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The ...
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This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The differential topology of links is a beautiful piece of mathematics, and the chapter offers a hands-on ‘user's guide’ approach with much emphasis on the famous work of Brieskorn in determining the difieomorphism types of certain homotopy spheres. This includes a presentation of the well known Brieskorn graph theorem as well as the geometry of Brieskorn-Pham links. When the singularities arise from weighted homogeneous polynomials, the links have a natural Sasakian structure with either definite (positive or negative) or null basic first Chern class. Emphasis is given to the positive case which corresponds to having positive Ricci curvature.Less

This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The differential topology of links is a beautiful piece of mathematics, and the chapter offers a hands-on ‘user's guide’ approach with much emphasis on the famous work of Brieskorn in determining the difieomorphism types of certain homotopy spheres. This includes a presentation of the well known Brieskorn graph theorem as well as the geometry of Brieskorn-Pham links. When the singularities arise from weighted homogeneous polynomials, the links have a natural Sasakian structure with either definite (positive or negative) or null basic first Chern class. Emphasis is given to the positive case which corresponds to having positive Ricci curvature.

*Clifford Henry Taubes*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0016
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter first examines some fundamental examples where the Riemann curvature tensor has an especially simple form. It then discusses the ways in which the Riemann and Ricci curvatures are used ...
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This chapter first examines some fundamental examples where the Riemann curvature tensor has an especially simple form. It then discusses the ways in which the Riemann and Ricci curvatures are used to study questions in geometry and differential topology. Topics covered include spherical metrics, flat metrics, and hyperbolic metrics; the Schwarzchild metric; curvature conditions; the Gauss-Bonnet formula; metrics on manifolds of dimension 2; conformal changes; sectional curvatures and universal covering spaces; the Jacobi field equation; constant sectional curvature and the Jacobi field equation; manifolds of dimension 3; and the Riemannian curvature of a compact matrix group.Less

This chapter first examines some fundamental examples where the Riemann curvature tensor has an especially simple form. It then discusses the ways in which the Riemann and Ricci curvatures are used to study questions in geometry and differential topology. Topics covered include spherical metrics, flat metrics, and hyperbolic metrics; the Schwarzchild metric; curvature conditions; the Gauss-Bonnet formula; metrics on manifolds of dimension 2; conformal changes; sectional curvatures and universal covering spaces; the Jacobi field equation; constant sectional curvature and the Jacobi field equation; manifolds of dimension 3; and the Riemannian curvature of a compact matrix group.

*Charles Fefferman and C. Robin Graham*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691153131
- eISBN:
- 9781400840588
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153131.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics ...
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This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics constructed in Theorems 3.7, 3.9 and 3.10 are the only formal expansions of metrics for ρ > 0 or ρ < 0 involving positive powers of ¦ ρ r ρ and log ¦ ρ r ρ, which are homogeneous of degree 2, Ricci-flat to infinite order, and in normal form. Convergence of formal series determined by Fuchsian problems such as these in the case of real-analytic data has been considered by several authors. In particular, results of [BaoG] can be applied to establish the convergence of the series occurring in Theorems 3.7 and 3.9 (and also in Theorem 3.10 if the obstruction tensor vanishes) if g and h are real-analytic. Convergence results including also the case when log terms occur in Theorem 3.10 are contained in [K].Less

This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics constructed in Theorems 3.7, 3.9 and 3.10 are the only formal expansions of metrics for ρ > 0 or ρ < 0 involving positive powers of ¦ ρ r ρ and log ¦ ρ r ρ, which are homogeneous of degree 2, Ricci-flat to infinite order, and in normal form. Convergence of formal series determined by Fuchsian problems such as these in the case of real-analytic data has been considered by several authors. In particular, results of [BaoG] can be applied to establish the convergence of the series occurring in Theorems 3.7 and 3.9 (and also in Theorem 3.10 if the obstruction tensor vanishes) if g and h are real-analytic. Convergence results including also the case when log terms occur in Theorem 3.10 are contained in [K].