By Jacqueline Lelong-Ferrand

ISBN-10: 2130388515

ISBN-13: 9782130388517

Approche axiomatique de los angeles géométrie

* through los angeles constitution vectorielle

* comme constitution d'incidence (à los angeles façon de Veblen & younger)

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T) = FlLX (t, x) , or FlL = µα(t) , for all t . x = Fl (t, x) , or Flt = µα(t) , for all t. α(s) = Proof. (1) =⇒ (4). We have dt d d | µ α(s) = T (µ ). α(t)). α(t) = Fl (t, x). (4) =⇒ (2). This is clear. (2) =⇒ (1). We have d ds α(t)α(s) = d ds (µα(t) α(s)) d = T (µα(t) ) ds α(s) = T (µα(t) )LX (α(s)) = LX (α(t)α(s)) LX X and α(t)α(0) = α(t). So we get α(t)α(s) = FlLX (s, α(t)) = FlL s Flt (e) = FlLX (t + s, e) = α(t + s). (4) ⇐⇒ (5). 14). α(−t). y. (5) =⇒ (3) =⇒ (1) can be shown in a similar way.

A1n .. . Aj−1 n 0 Aj+1 n .. . Ann Prove that C(A)A = AC(A) = det(A) · I (Cramer’s rule)! This can be done by remembering the the expansion formula for the determinant during multiplying it out. Prove that d(det)(A)X = Trace(C(A)X)! There are two ways to do this. The first one is to check that the standard inner product on L(Rn , Rn ) is given by A, X = Trace(A⊤ X), and by computing the gradient of det at A. 19): det(A + t Id) = tn + tn−1 Trace(A) + tn−2 cn2 (A) + · · · + t cnn−1 (A) + det(A).

Draft from April 18, 2007 Peter W. Michor, 32 Chapter I. 25 Proof. (1) =⇒ (2). Let X ∈ XE and let L be the leaf through x ∈ M , with i∗ X i : L → M the inclusion. TF li∗ X (−t,x) L = EF lX (−t,x) . ∗ This implies that (FlX t ) Y ∈ XE for any Y ∈ XE . (2) =⇒ (4). In fact (2) says that XE ⊂ aut(E). ∗ (4) =⇒ (3). We can choose W = aut(E) ∩ XE : for X, Y ∈ W we have (FlX t ) Y ∈ XE ; so W ⊂ S(W) ⊂ XE and E is spanned by W. (3) =⇒ (1). We have to show that each point x ∈ M is contained in some integral submanifold for the distribution E.

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