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Unformatted text preview: n = {(x0 , . . . , xn ) ∈ Rn+1 , −x2 + x2 + . . . + x2 = −1, x0 > 0}
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n
with Iso(X ) equal to the subgroup O(n, 1)+ of index 2 of the orthogonal
group O(n, 1) which consists of transformations of spinor norm 1, that is,
transformations that can be written as a product of reﬂections in vectors
with positive norm. The Riemannian metric is induced by the hyperbolic
metric in Rn+1 ,
ds2 = −dx2 + dx2 + . . . + dx2 .
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n
We will be using a projective model of H n , considering H n as the image of the
subset
C = {(x0 , . . . , xn ) ∈ Rn+1 , −x2 + x2 + . . . + x2 < 0}
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n REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 7 in the projective space Pn (R). The isometry group of the projective model is
naturally identiﬁed with the group PO(n, 1). By choosing a representative of a
point from H n with x0 = 1, we can identify H n with the real ball
K n : {(x1 , . . . , xn ) : x2 = x2 + . . . + x2 < 1}
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n
(the Klein model ). The metric is given by
ds2 = 1
1 − x2 n n dx2 +
i
i=1 1
(
xi dxi )2 .
(1 − x2 )2 i=1 The closure of H n in Pn (R) is equal to the image of the set
¯
C = {(x0 , . . . , xn ) ∈ Rn+1 , −x2 + x2 + . . . + x2 ≤ 0}
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n
in Pn (R). The boundary is called the absolute.
One deﬁnes the notion of a hyperplane in a space of constant curvature. If
X = E n , a hyperplane is an aﬃne hyperplane. If X = S n , a hyperplane is the
intersection of S n with a linear hyperplane in Rn+1 (a great circle when n = 2).
If X = H n , a hyperplane is the nonempty intersection of H n with a projective
hyperplane in Pn (R).
Each hyperplane H in E n is a translate a + L = {x + a, x ∈ L} of a unique
˜
linear hyperplane H in the corresponding standard euclidean space V = Rn . If
X n = S n or H n , then a hyperplane H is uniquely deﬁned by a linear hyperplane
˜
H in V = Rn+1 equipped with the standard symmetric bilinear form of Sylvester
signature (t+ , t− ) = (n + 1, 0) or (n, 1).
Any point x ∈ V can be written uniquely in the form
x = h + v,
˜
where h ∈ H and v ∈ V is orthogonal to H . We deﬁne a reﬂection with mirror
hyperplane H by the formula
rH (x) = h − v.
One can also give a uniform deﬁnition of a hyperplane in a space of constant
curvature as a totally geodesic hypersurface and deﬁne a reﬂection in such a space
as an isometric involution whose set of ﬁxed points is a hyperplane.
Let H be a hyperplane in X n . Its complement X n \ H consists of two connected
components. The closure of a component is called a halfspace. A reﬂection rH
permutes the two halfspaces. One can distinguish the two halfspaces by a choice of
one of the two unit vectors in V orthogonal to the corresponding linear hyperplane
˜
H . We choose it so that it belongs to the corresponding halfspace. For any vector v
perpendicular to a hyperplane in H n we have (v, v ) > 0 (otherwise the intersection
of the hyperplane with H n is empty).
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+
Let H1 , H2 be two halfspaces, and e1 , e2 be the corresponding unit vectors. If
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+
−
−
n
X = H ,...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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