Modular Forms course lecture notes 8

Modular Forms course lecture notes 8 - Spheres and...

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( April 16, 2011 ) Spheres and hyperbolic spaces Paul Garrett [email protected] http: / /www.math.umn.edu/ e garrett/ Basic examples of non-Euclidean geometries are best studied by studying the groups that preserve the geometries. In fact, rather than specifying the geometry, we specify the group . The group-invariant geometry on spheres is the familiar spherical geometry , with a simple relation to the ambient Euclidean geometry. The group-invariant geometry on real and complex n -balls is hyperbolic geometry, in the sense that there are infinitely many straight lines (geodesics) through a given point not on a given straight line, thus contravening the parallel postulate for Euclidean geometry. We will not directly consider geometric notions, since the transitive group action determines structure in a more useful form. Still, this explains the terminology. Rotations of spheres Holomorphic rotations Action of GL n +1 ( C ) on projective space P n Real hyperbolic n -space Complex hyperbolic n -space 1. Rotations of spheres The elementary ideas of this section are important enough to deserve a review. Let h , i be the usual inner product on R n , namely h x, y i = n X i =1 x i y i (where x = ( x 1 , . . . , x n ) and y = ( y 1 , . . . , y n )) The distance function is definable in terms of this, as usual, by distance from x to y = | x - y | (where | x | = h x, x i 1 / 2 ) The standard ( n - 1)-sphere S n - 1 in R n is S n - 1 = { x R n : | x | = 1 } As usual, the general linear and special linear groups of size n (over R ) are GL n ( R ) = { n -by- n invertible real matrices } = general linear group SL n ( R ) = { g GL n ( R ) : det g = 1 } = special linear group The modifier special refers to the determinant-one condition. Our definition of rotation [1] in R n will be a linear map of R n to itself which preserves distances , angles , and has determinant one (to preserve orientation ). The condition that a linear map g preserves angles and [1] A direct way to define rotation in R 2 is as a linear map with a matrix of the form cos θ sin θ - sin θ cos θ for some real θ . This definition is deficient insofar as it depends on a choice of basis. A definition in R 3 is that a rotation is a linear map g that has an axis , in the sense that there is a line L fixed by g , and on the orthogonal complement L of L the restriction of g is a two-dimensional rotation. For this to make sense, one must have understood that the two-dimensional definition is independent of basis, and that g does stabilize the orthogonal complement of any line fixed by it. Indeed, there is no necessity of reference here to R n . We could instead use any R vector space with an inner product. 1
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Paul Garrett: Spheres and hyperbolic spaces (April 16, 2011) distances is exactly that the inner product is preserved, in the sense that h gx, gy i = h x, y i Define the standard orthogonal group O n ( R ) = orthogonal group = angle-and-distance-preserving group = { g GL n ( R ) : h gx, gy i = h x, y i for all x, y R n } Since distances are preserved, O n ( R ) stabilizes the (unit) sphere S n - 1 in R n . The preservation of the inner product does not
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