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98 8 Continuous groups of transformationstangent to O p . It can be shown that the choice of base point p in O p doesnot affect (µ p ) ∗ . Hence, using a map (µ p ) ∗ in each O p , we can define a Liealgebra of vector fields on M by taking the image of the Lie algebra ofG. At the risk of some confusion we use {ξ A } to denote a basis of eitherLie algebra. The two algebras are isomorphic because G is assumed to beeffective, and so (µ p ) ∗ v = 0 for all p only if v = 0.The stability group of p is generated by those v such that (µ p ) ∗ v =0 atp; this is clearly the kernel of the map (µ p ) ∗ at q 0 . Denoting the dimensionof O p by d we thus haver = d + s. (8.20)The classical theorems on continuous transformation groups can beexpressed asTheorem 8.7 (Lie’s first fundamental theorem). An action µ : G×M →M of a continuous (Lie) group of transformations defines and is definedby a linear map of the right-invariant vector fields on G r onto an r-dimensional set of (smooth) vector fields on M.Theorem 8.8 (Lie’s second fundamental theorem). A set of r (smooth)linearly independent vector fields {ξ A } on M obeying (8.6) defines and isdefined by a continuous (Lie) group of transformations on M.A single generator ξ of a transformation group G r gives rise to a oneparametersubgroup Φ x (see §2.8) of G r , and by choosing one point pin each orbit of this group as x = 0 we can find a coordinate x in Msuch that ξ = ∂ x (the term trajectory is sometimes reserved for such onedimensionalorbits). If there are m commuting generators {ξ A } (formingan Abelian subgroup), all non-zero at p, then one can thus find m coordinates(x 1 ,...,x m ) such that ξ A = ∂/∂x A (A =1,...,m).8.4 Groups of motionsManifolds with structure, such as Riemannian manifolds V n , may admit(continuous) groups of transformations preserving this structure. In a V n ,the map Φ t corresponding (as in §2.8) to a conformal motion obeying(6.10) has the property (Φ t g) ab =e 2U g ab , where U is the integral of the φin (6.10) along a curve, i.e. it preserves the metric up to a factor. This isa conformal transformation (§3.7), whence the name conformal motion.It is a homothety if φ is constant, and a motion (or isometry), whosegenerator obeys Killing’s equation (6.11) and which preserves the metric,if φ = 0. Here we shall consider motions. Homothety groups are discussedfurther in §8.7 and more general symmetries in §35.4.