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Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771. Introduction to Smooth Manifolds, Springer-Verlag, Graduate Texts in Mathematics, 2002, 2nd edition 2012 [6] Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds.
Let :, (,) be a (left) group action of a Lie group on a smooth manifold ; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G {\displaystyle G} on M {\displaystyle M} consists of a Lie group homomorphism G → D i f f ( M ) {\displaystyle G\to \mathrm {Diff} (M)} .
Important to applications in mathematics and physics [1] is the notion of a flow on a manifold. In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to .
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.
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 ...
In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold. Congruences are an important concept in general relativity , and are also important in parts of Riemannian geometry .
Types of manifolds in engineering include: Exhaust manifold An engine part that collects the exhaust gases from multiple cylinders into one pipe. Also known as headers. Hydraulic manifold A component used to regulate fluid flow in a hydraulic system, thus controlling the transfer of power between actuators and pumps Inlet manifold (or "intake ...
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