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A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an -manifold with boundary is an ()-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold.
A Riemannian manifold (M, g) is said to be homogeneous if for every pair of points x and y in M, there is some isometry f of the Riemannian manifold sending x to y. This can be rephrased in the language of group actions as the requirement that the natural action of the isometry group is transitive. Every homogeneous Riemannian manifold is ...
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...
Tangent space. In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a ...
In the mathematical field of Riemannian geometry, 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.
Copper fittings for soldered joints. Pipe fittings: 1) Copper (solder); 2) Iron or brass (threaded); 3) Brass (compression); 4) Brass (compression to solder); 5) Brass adapters. PVC fittings. A fitting or adapter is used in pipe systems to connect sections of pipe (designated by nominal size, with greater tolerances of variance) or tube ...
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In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. [1] Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the ...
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