By B. N. Apanasov

This e-book offers the 1st systematic account of conformal geometry of n-manifolds, in addition to its Riemannian opposite numbers. A unifying subject matter is their discrete holonomy teams. particularly, hyperbolic manifolds, in measurement three and better, are addressed. The therapy covers additionally proper topology, algebra (including combinatorial workforce conception and forms of workforce representations), mathematics matters, and dynamics. growth in those parts has been very quick over the past twenty years, in particular as a result of Thurston geometrization application, resulting in the answer of many tough difficulties. a robust attempt has been made to indicate new connections and views within the box and to demonstrate quite a few points of the idea. An intuitive technique which emphasizes the tips in the back of the buildings is complemented by way of a great number of examples and figures which either use and help the reader's geometric mind's eye. The textual content could be of price to graduate scholars and researchers in topology, geometry, staff representations and theoretical physics.

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Put , M =U (X( = {(z, ()E C" XlPn-llzE n. Let (Zto ... , zn) be the standard coordinates on en, and «(to ... , (n) the homogeneous coordinates on lPn-I. In terms of these coordinates, we denote a point (z, () by (Zto ... , Zn, (to ... , (n). Then Z E ( if (jZk - Zj(k = 0 for j, k = I, ... , n. 18) j, k= I, ... , n. is covered by n pieces of coordinate neighbourhoods ~ = ... ,n,C"xlPn-1 is covered by en X Vis. To verify that M is a submanifold enxlPn-t, consider MI=MIIC"XVI. Let (W2, ••• , Wk, ••• , wn ) be the inhomogeneous coordinates on VI where Wk = (kill.

Z~(p)) with centre q. Take r>O such that the closed polydisk {(z~, ... , z~) Ilz~1 ~ r, k = 1, ... , n} is contained in the range of Zq, and put Ur(q) = {p Ilz~(p)1 < r, ... , Iz~(p)1 < r}. Given q E W, if we choose r sufficiently small, we have g( Ur(q)) n Ur(q) = 0 for any g E G except the identity. In fact, if otherwise, there is an element gn E G, gn ¥- 1, for each n = 1,2, ... , such that gn( Un) nUn ¥- 0, where Un = Ur/n(q). Then gn(U,)n U,¥-0 for any n. Since [U,] is compact, and Gis properly discontinuous, {g" ...

If Tjk is biholomorphic for any j, k such that ~ n Uk ~ 0, each Zj: p ~ Zj (p) is called local complex coordinates defined on ~, and the collection {z" ... , Zj, ... } is called a system of local complex coordinates on L. 1. If a system of local complex coordinates {z" ... , Zj, ... 1. Complex Manifolds is defined on :l. A connected Hausdorff space is called a complex manifold if a complex structure is defined on it. We denote a complex manifold by the letters M, N, etc. The system oflocal complex coordinates {Zh ...