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Ahlfors, Lars V. (1961), "Some remarks on Teichmüller's space of Riemann surfaces", Annals of Mathematics, Second Series, 74 (1): 171– 191, doi:10.2307/1970309 ...
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.
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.
Cartan connection. Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.. Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to R n via the exponential map; for metric spaces, the statement that a connected, simply connected complete ...
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 ...
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 ...
The fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-) Riemannian connection of the given metric.