*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/9780198564959.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter begins by introducing various geometries that play important roles in the way they relate to Sasakian structures. It espouses the point of view that a geometric structure is best ...
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This chapter begins by introducing various geometries that play important roles in the way they relate to Sasakian structures. It espouses the point of view that a geometric structure is best described as a G-structure which may or may not be (partially) integrable. Some selected topics include: Riemannian metrics, complex structures, symplectic structures, contact structures, quaternionic structures, group actions, pseudogroups, sheaves, bundles, connections, holonomy, curvature and integrability.Less

This chapter begins by introducing various geometries that play important roles in the way they relate to Sasakian structures. It espouses the point of view that a geometric structure is best described as a G-structure which may or may not be (partially) integrable. Some selected topics include: Riemannian metrics, complex structures, symplectic structures, contact structures, quaternionic structures, group actions, pseudogroups, sheaves, bundles, connections, holonomy, curvature and integrability.

*Andrea Braides*

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.003.0004
- Subject:
- Mathematics, Applied Mathematics

Homogenization problems for a general class of integrals are solved by a direct approach. Different homogenization formulas are given, both in an asymptotic form and as a cell problem (in the convex ...
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Homogenization problems for a general class of integrals are solved by a direct approach. Different homogenization formulas are given, both in an asymptotic form and as a cell problem (in the convex case). These are applied in the study of the asymptotic behaviour of Riemannian metrics and Hamilton-Jacobi equations.Less

Homogenization problems for a general class of integrals are solved by a direct approach. Different homogenization formulas are given, both in an asymptotic form and as a cell problem (in the convex case). These are applied in the study of the asymptotic behaviour of Riemannian metrics and Hamilton-Jacobi equations.

*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/9780199605880.003.0007
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter discusses the following: metrics and transition functions for real vector bundles; metrics and transition functions for complex vector bundles; metrics, algebra and maps; and a metric on ...
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This chapter discusses the following: metrics and transition functions for real vector bundles; metrics and transition functions for complex vector bundles; metrics, algebra and maps; and a metric on the tangent bundle of a manifold M (TM), called a Riemannian metric.Less

This chapter discusses the following: metrics and transition functions for real vector bundles; metrics and transition functions for complex vector bundles; metrics, algebra and maps; and a metric on the tangent bundle of a manifold M (TM), called a Riemannian metric.

*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/9780199605880.003.0008
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

Let M denote a smooth manifold. A metric on TM can be used to define a notion of the distance between any two points in M and the distance travelled along any given path in M. This chapter first ...
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Let M denote a smooth manifold. A metric on TM can be used to define a notion of the distance between any two points in M and the distance travelled along any given path in M. This chapter first explains how this is done then considers the distance minimizing paths. The discussions cover Riemannian metrics and distance; length minimizing curves; the existence of geodesics; examples of metrics with their corresponding geodesics; geodesics on SO(n); geodesics on U(n) and SU(n); and geodesics and matrix groups.Less

Let M denote a smooth manifold. A metric on TM can be used to define a notion of the distance between any two points in M and the distance travelled along any given path in M. This chapter first explains how this is done then considers the distance minimizing paths. The discussions cover Riemannian metrics and distance; length minimizing curves; the existence of geodesics; examples of metrics with their corresponding geodesics; geodesics on SO(n); geodesics on U(n) and SU(n); and geodesics and matrix groups.