By Kolar, Michor, Slovak.

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Within the Spring of 1966, I gave a sequence of lectures within the Princeton college division of Physics, aimed toward fresh mathematical leads to mechanics, particularly the paintings of Kolmogorov, Arnold, and Moser and its software to Laplace's query of balance of the sun approach. Mr. Marsden's notes of the lectures, with a few revision and growth through either one of us, grew to become this publication.

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Example. The general linear group GL(n, R) is the group of all invertible real n × n-matrices. It is an open subset of L(Rn , Rn ), given by det = 0 and a Lie group. Similarly GL(n, C), the group of invertible complex n × n-matrices, is a Lie group; also GL(n, H), the group of all invertible quaternionic n × n-matrices, is a Lie group, but the quaternionic determinant is a more subtle instrument here. 4. 5. Example. The orthogonal group O(n, R) is the group of all linear isometries of (Rn , , ), where , is the standard positive definite inner product on Rn .

An integral curve of X is a smooth curve c : I → M with c(t) ˙ = X(t, c(t)) for all t ∈ I, where I is a subinterval of J. ¯ ∈ X(J × M ), given by X(t, ¯ x) = There is an associated vector field X (1t , X(t, x)) ∈ Tt R × Tx M . By the evolution operator of X we mean the mapping ΦX : J × J × M → M , defined in a maximal open neighborhood of the diagonal in M ×M and satisfying the differential equation d X dt Φ (t, s, x) X = X(t, ΦX (t, s, x)) Φ (s, s, x) = x. 7), where ΦX t,s (x) = Φ(t, s, x). 30 Chapter I.

Then O(n, R) = f −1 (In ); so O(n, R) is closed. Since it is also bounded, O(n, R) is compact. X t , so ker df (In ) = {X : X + X t = 0} is the space o(n, R) of all skew symmetric n × n-matrices. Note that dim o(n, R) = 21 (n − 1)n. (A−1 )t . The mapping f takes values in Lsym (Rn , Rn ), the space of all symmetric n × n-matrices, and dim ker df (A) + dim Lsym (Rn , Rn ) = 1 1 2 n n n n 2 (n − 1)n + 2 n(n + 1) = n = dim L(R , R ), so f : GL(n, R) → Lsym (R , R ) −1 is a submersion. 10 that O(n, R) is a submanifold of GL(n, R).