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A moving frame on a submanifold M of G/H is a section of the pullback of the tautological bundle to M. Intrinsically a moving frame can be defined on a principal bundle P over a manifold. In this case, a moving frame is given by a G-equivariant mapping φ : P → G, thus framing the manifold by elements of the Lie group G.
9 KINEMATICS OF MOVING FRAMES 71 This result is often written in terms of body-referenced velocity v: ∂ r v = ω × r + + vo, ∂t where vo is the body-referenced velocity of the origin. The total velocity of the particle is equal to the velocity of the reference frame origin, plus a component due to rotation of this frame.
2. Equivariant Moving Frames. We begin by describing the general equivariant moving frame construction. Let Gbe an r-dimensional Lie group acting smoothly on an m-dimensional manifold M. Definition 2.1. A moving frame is a smooth, G-equivariant map ρ:M→ G. There are two principal types of equivariance: ρ(g·z) = ˆ g·ρ(z) left moving frame
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geometry in terms of moving frames. Our indices shall have the range i;j;k;:::= 1;2 and A;B;C;:::= 1;2;3 and we follow the convention that sums are over repeated indices. Let SˆR3 be a surface and let the dot product of R3 be given h;i. Let a local chart for Sbe given by the map X: U!Swhere UˆR2 is an open set.
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Learn about the new, algorithmic theory of moving frames, a powerful tool for studying geometric properties and invariants under transformation groups. The paper surveys applications in geometry, computer vision, invariant theory, and numerical analysis.
ing frames where the use of differential forms is emphasized[when compared with tensor fields, differential forms have the advantage that they can be pulled-back via smooth maps, and we have the powerful tool of exterior derivative]. 1. Cartan’s method of moving frames ¶ The connection 1-forms for a linear connection in a local frame.
Abstract. This chapter presents the equivariant method of moving frames for finite-dimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. Introduction. According to Akivis [1], the method of moving ...
