By Ibragimov N.
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This complicates the problem to a great extent. 2. e. such that there exists a nontrivial local group of conformal transformations in a neighborhood of every point x ∈ V4 . In particular, I think that if V4 is a topologically 1-connected space with locally nontrivial conformal group, then the dimension of the maximal conformal group is the same at any point in V4 . Clarify if this hypothesis is correct or false. 1: LIE GROUPS IN MATHEMATICAL PHYSICS (1972) 51 Chapter 3 Group analysis of second-order equations This chapter is dedicated to investigation of symmetry properties of partial differential equations of the second order.
N). ∂a a=0 It is assumed here that the expressions for the quantities y¯ij are obtained from Eqs. 6) via xi , yij and the group parameter a. Let us assume that these expressions are substituted into Eqs. 6). 7) and the conditions y¯ij |a=0 = yij we obtain ηij = ηkl δik δjl + ykl l ∂ξ k l ∂ξ k ∂ξ k k ∂ξ + y δ δ = η + y + y = 0, kl ij kj ik i ∂xi j ∂xj ∂xi ∂xj whence ηij = − yik ∂ξ k ∂ξ k + y kj ∂xj ∂xi (i, j = 1, . . , n). 8) implies that if X and Y are generators of two oneparameter groups, then [X Y ] = [X, Y ].
5. 1) is invariant with respect to the extended group G. Proof. The necessary and sufficient condition of invariance of the manifold G with respect to the group G consists in satisfaction of equations X(gij (x) − yij )|G = 0 (i, j = 1, . . 8) of the group G. 4). 2 Generalized motions. Defect Let us consider the general situation. 1) in the space Vn . Let us check how the corresponding manifold G varies with transformations of the extended group G. Every transformation T ∈ G leads to the manifold T (G) (see § 3) equivalent to the manifold G.