33 fundamental property the theorem for the product

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . Example of G final parametric manifold G final parametric manifold is visualised by a 2D surface in our 3D space for a particular case, too simple to be of practical use, but one that allows the general case of a dimension n rank 3 manifold to be pictured. Given the Ω perturbation as the most common for an object, reflected by a Ginit metric tensor with a dimension n = 2 and rank 1 ( b 2 − a ⋅ c = 0 ), with a b Ginit = b c 28 P. Serré A. Rivière and A. Clé , ment the G final manifold will then represent all 2-dimension, rank 1 tensors. X G final = Y q2 a b 1 + q1 q3 Y 1 + q1 ⋅ = ⋅ Z q3 1 + q4 b c q2 1 + q4 By identifying the 2 members of the equation, we obtain a parametric representation (see Figure 1) with a 4-parameter manifold: X = a + 2 ⋅ a ⋅ q1 + 2 ⋅ b ⋅ q2 + 2 ⋅ b ⋅ q1 ⋅ q2 + a ⋅ q12 + c ⋅ q2 2 Y = b + a ⋅ q3 + b ⋅ q1 + b ⋅ q4 + c ⋅ q2 + a ⋅ q1 ⋅ q3 + b ⋅ q1 ⋅ q4 + b ⋅ q2 ⋅ q3 + c ⋅ q2 ⋅ q4 2 2 Z = c + 2 ⋅ b ⋅ q3 + 2 ⋅ c ⋅ q4 + 2 ⋅ b ⋅ q3 ⋅ q4 + a ⋅ q3 + c ⋅ q4 This is the...
View Full Document

Ask a homework question - tutors are online