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  2. Riemannian manifold - Wikipedia

    en.wikipedia.org/wiki/Riemannian_manifold

    A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.

  3. Geometric analysis - Wikipedia

    en.wikipedia.org/wiki/Geometric_analysis

    The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem (1st ed.). Springer. ISBN 978-3-642-16285-5. Jost, Jürgen (2005). Riemannian geometry and Geometric Analysis (4th ed.). Springer. ISBN 978-3-540-25907-7. Lee, Jeffrey M. (2009). Manifolds and Differential Geometry. American Mathematical ...

  4. Sage Manifolds - Wikipedia

    en.wikipedia.org/wiki/Sage_Manifolds

    An important class of treated manifolds is that of pseudo-Riemannian manifolds, among which Riemannian manifolds and Lorentzian manifolds, with applications to General Relativity. In particular, SageManifolds implements the computation of the Riemann curvature tensor and associated objects ( Ricci tensor , Weyl tensor ).

  5. Riemannian geometry - Wikipedia

    en.wikipedia.org/wiki/Riemannian_geometry

    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.

  6. Hopf conjecture - Wikipedia

    en.wikipedia.org/wiki/Hopf_conjecture

    The Hopf conjecture is an open problem in global Riemannian geometry. It goes back to questions of Heinz Hopf from 1931. A modern formulation is: A compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic.

  7. Calibrated geometry - Wikipedia

    en.wikipedia.org/wiki/Calibrated_geometry

    In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration, meaning that: φ is closed: dφ = 0, where d is the exterior derivative

  8. Maps of manifolds - Wikipedia

    en.wikipedia.org/wiki/Maps_of_manifolds

    Just as there are various types of manifolds, there are various types of maps of manifolds. PDIFF serves to relate DIFF and PL, and it is equivalent to PL.. In geometric topology, the basic types of maps correspond to various categories of manifolds: DIFF for smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and TOP for ...

  9. Normal coordinates - Wikipedia

    en.wikipedia.org/wiki/Normal_coordinates

    A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter).

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