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Geometries of Homogeneous Spaces 1. Rotations of Spheres

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Geometries of Homogeneous Spaces 1. Rotations of Spheres (October 9, 2013) Geometries of homogeneous spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/mfms/notes 2013-14/06 homogeneous geometries.pdf] Basic examples of non-Euclidean geometries are best studied by studying the groups that preserve the geometries. 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, also rotation-invariant. The group-invariant geometry on real and complex n-balls is hyperbolic geometry: there are infinitely many straight lines (geodesics) through a given point not on a given straight line, contravening in surplus the parallel postulate for Euclidean geometry. 1. Rotations of spheres 2. Holomorphic rotations n 3. Action of GLn+1(C) on projective space P 4. Real hyperbolic n-space 5. Complex hyperbolic n-space 1. Rotations of spheres Let h; i be the usual inner product on Rn, namely n X hx; yi = xi yi (where x = (x1; : : : ; xn) and y = (y1; : : : ; yn)) i=1 The distance function is expressed in terms of this, as usual: distance from x to y = jx − yj (where jxj = hx; xi1=2) The standard (n − 1)-sphere Sn−1 in Rn is n−1 n S = fx 2 R : jxj = 1g The usual general linear and special linear groups of size n (over R) are 8 < GLn(R) = fn-by-n invertible real matricesg = general linear group : SLn(R) = fg 2 GLn(R) : det g = 1g = special linear group The modifier special refers to the determinant-one condition. [1] n n Our definition of rotation in R is a linear map on R preserving distances, angles, and with determinant one (to preserve orientation). To preserve angles and distances is exactly that the inner product is preserved, in the sense that hgx; gyi = hx; yi cos θ sin θ [1] A direct way to define rotation in 2 is as a linear map with a matrix of the form for some real R − sin θ cos θ 3 θ. This definition is deficient as it depends on a choice of basis. A definition in R is that a rotation is a linear map g with an axis, 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 to n refer to R . Any R vector space with an inner product works as well. 1 Paul Garrett: Geometries of homogeneous spaces (October 9, 2013) The standard orthogonal group is On(R) = orthogonal group = angle-and-distance-preserving group n = fg 2 GLn(R): hgx; gyi = hx; yi for all x; y 2 R g n−1 n Since distances are preserved, On(R) stabilizes the unit sphere S in R , and every sphere centered at 0. Preservation of the inner product does not fully distinguish rotations, since it does not imply the orientation- preserving (determinant-one) condition, which has to be added explicitly. Thus, the standard [2] special orthogonal group [3] , meant to be the group of rotations, is SOn(R) = special orthogonal group = rotation group = fg 2 On(R) : det g = 1g A common, expedient, but unenlightening, definition of the standard orthogonal group is > > On(R) = fg 2 GLn(R): g g = 1ng (g is g-transpose, 1n is the n-by-n identity) and the standard special orthogonal group is still SOn(R) = fg 2 On(R) : det g = 1g Fortunately, these two characterizations specify the same objects: [1.0.1] Claim: The two characterizations of orthogonal group are the same. That is, n > fg 2 GLn(R): hgx; gyi = hx; yi for all x; y 2 R g = fg 2 GLn(R): g g = 1ng Proof: The usual inner product is also expressible as X > > hx; yi = xiyi = y x (for column vectors, y = y-transpose) i > For g g = 1n, compute directly hgx; gyi = (gy)>(gx) = y> (g>g) x = y>x = hx; yi > from which g g = 1n implies h; i is preserved. On the other hand, suppose g preserves h; i. The main trick is that, for column vectors v1; : : : ; vn and w1; : : : ; wn, all of length n, inserted as the columns of two matrices V and W , > > th W V = [ w1 : : : wn ] [ v1 : : : vn ] = n-by-n matrix with (i; j) entry hwi; vji Running this backward, for g 2 GLn(R) [2] In fact, the form of these definitions did not use the explicit form of the inner product, so applies as well to other possible inner products, as well. While in the near future we care mostly about the standard one, it is wise to present things in a form which does not needlessly depend on irrelevant particulars. [3] As with the special linear group, the modifier special on special orthogonal group refers to the determinant-one condition. 2 Paul Garrett: Geometries of homogeneous spaces (October 9, 2013) th > n-by-n matrix with (i; j) entry hgwi; gvji = [ gw1 : : : gwn ] [ gv1 : : : gvn ] = (gW )>(gV ) = W > (g>g) V For g preserving inner products, slightly cleverly taking v1; : : : ; vn and w1; : : : ; wn to be the standard basis e1; : : : ; en, the previous relation becomes th > > > > > n-by-n matrix with (i; j) entry hgei; geji = W (g g) V = 1n (g g) 1n = g g > proving that g g = 1n. Thus, the two characterizations of orthogonal groups agree. === > Regarding possible values of determinants of elements of On(R), observe that for any g such that g g = 1 2 > > (det g) = det g · det g = det(g g) = det 1n = 1 so det g = ±1. n−1 [1.0.2] Claim: The action of SOn(R) on S is transitive, for n ≥ 2. n−1 Proof: First show that, given x 2 S , there is g 2 On(R) such that ge1 = x, where e1; : : : ; en is the n standard basis for R . That is, construct g 2 On(R) such that the left column of g is x. Indeed, complete n [4] x to an R-basis x; x2; x3; : : : ; xn for R . Then apply the Gram-Schmidt process to find an orthonormal n (with respect to the standard inner product) basis x; v2; : : : ; vn for R . As observed in the previous proof, > the condition g g = 1n, is exactly the assertion that the columns of g form an orthonormal basis. Thus, taking x; v2; : : : ; vn as the columns of g gives g 2 On(R) such that ge1 = x. As noted above, the determinant of this g is ±1. To ensure that it is 1, replace vn by −vn if necessary. This still gives ge1 = x, giving the transitivity. === [1.0.3] Claim: The isotropy group SOn(R)en of the last standard basis vector en = (0;:::; 0; 1) is A 0 (isotropy group) = SO ( ) = f : A 2 SO ( )g ≈ SO ( ) n R en 0 1 n−1 R n−1 R Thus, by transitivity, as SOn(R)-spaces n−1 S ≈ SOn(R) = SOn−1(R) Proof: There are at least two ways to think about this. First, think of the geometry specified by h; i. Second, manipulate matrices. Taking the first course first, claim that for g fixing en, the orthogonal complement ? n en = fv 2 R : hv; eni = 0g ? ? is stabilized by g, and on en the linear map preserves (the restriction to en of) h; i. Indeed, for hv; eni = 0 and gen = en, hgv; eni = hgv; geni = hv; eni = 0 since g preserves the inner product, showing that g stabilizes the orthogonal complement to en. Certainly g ? n preserves the restriction to en of h; i, since it preserved h; i on the whole space R . And, on the other hand, [4] Recall that, given a basis v1; : : : ; vn for a (real or complex) vector space with an inner product (real-symmetric or complex hermitian), the Gram-Schmidt process produces an orthogonal or orthonormal basis, as follows. Replace v1 by v1=jv1j to give it length 1. Then replace v2 first by v2 − hv2; v1iv1 to make it orthogonal to v1 and then by v2=jv2j to give it length 1. Then replace v3 first by v3 − hv3; v1iv1 to make it orthogonal to v1, then by v3 − hv3; v2iv2 to make it orthogonal to v2, and then by v3=jv3j to give it length 1. And so on. 3 Paul Garrett: Geometries of homogeneous spaces (October 9, 2013) ? ? for g stabilizing en and preserving the restriction of h; i to vo , define (an extension of) g on en by gen = en. 0 0 0 0 ? To check that this extended g is in On(R), for general vectors v = v +aen and w = w +ben with v ; w 2 en , a; b 2 R, a natural computation gives 0 0 0 0 0 0 hgv; gwi = hg(v + aen); g(w + beni = hgv + agen; gw + bgeni = hgv + aen; gw + beni 0 0 0 0 0 0 = hgv ; gw i + haen; beni = hv ; w i + haen; beni = hv + aen; w + beni = hv; wi Thus, the extension does preserve h; i on the larger space, and the stabilizer subgroup of en is proven to be a copy of SOn−1(R). At heart, the matrix argument does the same things, but mutely. Let A b g = 2 O ( ) c d n R > be a block decomposition with A of size n − 1, etc., with g g = 1n and gen = en.
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