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By Yu B.J., Xu M.

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Now, the conditions on the function f , together with the continuity of A, guarantee that A(s)f (s) ds is close to A(0) = I, and hence is invertible. 8) A(t) = B(t) A(s)f (s)ds . Since B (t) is smooth and A(s)f (s)ds is just a constant matrix, this shows that A (t) is smooth. Now that A(t) is known to be differentiable, we may define X= d dt t=0 A(t). Our goal is to show that A(t) = etX . t2 . It follows that for each fixed t, A t m t mX +O 1 m2 = I+ t mX +O =I+ . Then, since A is a one-parameter group A(t) = A t m m 1 m2 m .

Then U ′ is open and the exponential takes U ′ homeomorphically onto V . 25. 23, then the inverse map exp−1 : V → g is called the logarithm for G. 26. 16) A = eX1 eX2 · · · eXn for some X1 , X2 , · · · Xn in g. Proof. Recall that for us, saying G is connected means that G is pathconnected. This certainly means that G is connected in the usual topological sense, namely, the only non-empty subset of G that is both open and closed is G itself. 16). In light of the Proposition, E contains a neighborhood V of the identity.

The Euclidean and Poincar´ e groups. Recall that the Euclidean group E(n) is (or can be thought of as) the group of (n + 1) × (n + 1) real matrices of the form   x1  ..   R .     xn  0 ··· 0 1 with R ∈ O(n). Now if X is an (n + 1) × (n + 1) real matrix such that etX is in d etX must be zero along the bottom row: E(n) for all t, then X = dt t=0   y1  ..   Y . 14) are actually in the Lie algebra of the Euclidean group. A simple computation shows that for n≥1  n   y1  ..   Yn Y n−1 y   Y .

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