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Toggle Differential geometry of curves and surfaces subsection. ... 3.6 Curvature of Riemannian manifolds. 3.7 Theorems in Riemannian geometry. 3.8 Isometry.
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics . For categorical listings see Category:Manifolds and its subcategories.
In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X.It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form.
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus , integral calculus , linear algebra and multilinear algebra .
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.
In each local chart a Riemannian metric is given by smoothly assigning a 2×2 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to the Jacobian matrix of the coordinate change. The manifold then has the structure of a 2-dimensional Riemannian manifold.
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 ...
Two theorems in the mathematical field of Riemannian geometry bear the name Myers–Steenrod theorem, both from a 1939 paper by Myers and Steenrod.The first states that every distance-preserving surjective map (that is, an isometry of metric spaces) between two connected Riemannian manifolds is a smooth isometry of Riemannian manifolds.
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