By Alfred Barnard Basset
This quantity is made from electronic pictures from the Cornell college Library ancient arithmetic Monographs assortment.
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We say that E is a smooth distribution if XE spans E. Note that every subset W ⊂ Xloc (M ) spans a distribution denoted by E(W), which is obviously smooth (the linear span of the empty set is the vector space 0). From now on we will consider only smooth distributions. Draft from September 15, 2004 Peter W. Michor, 36 Chapter I. 9)) such that Tx i(Tx N ) = Ei(x) for all x ∈ N . 13)), so that we need not specify the injective immersion i. An integral manifold of E is called maximal, if it is not contained in any strictly larger integral manifold of E.
Let ϕ : G → H be a smooth homomorphism of Lie groups. Then ϕ := Te ϕ : g = Te G → h = Te H is a Lie algebra homomorphism. 21), we shall see that any continuous homomorphism between Lie groups is automatically smooth. Proof. X = Lϕ (X) (ϕ(x)). So LX is ϕ-related to Lϕ (X) . 10) the field [LX , LY ] = L[X,Y ] is ϕ-related to [Lϕ (X) , Lϕ (Y ) ] = L[ϕ (X),ϕ (Y )] . So we have T ϕ ◦ L[X,Y ] = L[ϕ (X),ϕ (Y )] ◦ ϕ. If we evaluate this at e the result follows. Now we will determine the Lie algebras of all the examples given above.
Example. The group Sp(n). Let H be the division algebra of quaternions. We will use the following description of quaternions: Let (R3 , , , ∆) be the oriented Euclidean space of dimension 3, where ∆ is a determinant function with value 1 on a positive oriented orthonormal basis. The vector product on R 3 is then given by X × Y, Z = ∆(X, Y, Z). Now we let H := R3 × R, equipped with the following product: (X, s)(Y, t) := (X × Y + sY + tX, st − X, Y ). Now we take a positively oriented orthonormal basis of R3 , call it (i, j, k), and indentify (0, 1) with 1.
A treatise on the geometry of surfaces by Alfred Barnard Basset