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By Werner Greub, Stephen Halperin, Ray Vanstone

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E; F ) . k terms For each U E U , ~ ( ~ ) is( aa )symmetric k-linear map of E x x E into F. If all derivatives of p exist, p is called injinitely dzjfmentiable, or smooth. --+ V between open subsets U C E and V C F is called a diffeomorphism if it has a smooth inverse. Assume now that p: U + F is a map with a continuous derivative such that for some point a E U - - a E ”-F is a linear isomorphism. Then the inverse function theorem states that there are neighbourhoods U of a and V of p(a) such that p restricts to a diffeomorphism U% V .

D. Corollary: If {Ua I (Y E j } is any open covering of M, there is a a(i) of Jf into 9 such that partition of unity {f i 1 i E $1 and a map i I--+ carr fiC Ua(o, i E f . 9. Functiongerms. Let a be a fixed point of M. Two members f , g of Y ( M ) will be called a-equiwalent, f 7g , if and only if there is a neighbourhood U of a such that f (x) = g ( x ) , x E U . T h e equivalence classes so obtained are called function germs at a. We write Via for the germ represented by f E Y ( M )and Y a ( M )for the set of function germs at a.

Then f is smooth and a suitable g is given by = f(p f ( t - a> - t ) + f ( t - a) 9. The Cayley map: Let E be a real or complex vector space of finite dimension. Then 0 = {uELE j det(c + u) # 0) is an open submanifold of the vector space LE containing 0 E LE. We will show that the Cayley map f : 0 -+ LE given by f(u) = (1 - u)(c + uE u)-1 0 is an involution of 0. In fact, f(u) + 1 = (c - u)(c + u)-1 + + u)(c + (I u)-1 = 2(c + whence f :0 -+ 0. ))-' . f(f(4); is clearly smooth, it is a diffeomorphism.

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