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Download as PDF; Printable version; ... and hence a Riemannian metric. ... Scott A. (2010), Families of Riemann Surfaces and Weil-Petersson Geometry, CBMS Reg. Conf ...
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume.
Riemannian Geometry (PDF). Princeton: Princeton University Press. OCLC 5836010. Eisenhart, Luther Pfahler (1939). Coordinate Geometry. Dover Publishing. [7] Eisenhart, Luther Pfahler (1927). Non-Riemannian geometry (PDF). New York: American Mathematical Society. [8] Eisenhart, Luther Pfahler (1909). A treatise on the differential geometry of ...
Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i.e. a choice of positive-definite quadratic form on a manifold's tangent spaces which varies smoothly from point to point. This gives in particular local ideas of angle, length of curves, and volume.
With Richard Bishop, he applied his submersion calculations to the geometry of warped products, in addition to studying the fundamental role of convex functions and convex sets in Riemannian geometry, and for the geometry of negative sectional curvature in particular. An article with his former Ph.D. student Patrick Eberlein made a number of ...
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds.
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