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Download Geometry and Spectra of Compact Riemann Surfaces (Modern by Peter Buser PDF

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By Peter Buser

This monograph is a self-contained creation to the geometry of Riemann Surfaces of continuous curvature –1 and their size and eigenvalue spectra. It specializes in topics: the geometric concept of compact Riemann surfaces of genus more than one, and the connection of the Laplace operator with the geometry of such surfaces. examine employees and graduate scholars drawn to compact Riemann surfaces will locate the following a couple of necessary instruments and insights to use to their investigations.

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3. Namely: let p be an intersection point, let y c S = D be a lift of y and letp e y be a lift of p. (y) intersects y under some positive angle atp as shown in Fig. 1. Lift the homotopy between y and c to D. 5). l, §6] 25 Closed Geodesies implies that c intersects R(c). Since c is simple, the unique lifting property implies that for some constant co, R(C(T)) = C{T + co), a contradiction. 7. (iv). Since c is not smooth, y n c = 0 , by (ii). By (iii), y is simple. The proof that y and c bound an embedded annulus is of purely topological nature and is postponed to the Appendix (Proposition A.

3 except that we now admit orientation reversing overlap maps. ) If we interpret (1) as the definition of a curve in Fy then y is the 2-fold iterate of another closed geodesic. Curve c in Fig. 3 is the example of a simple closed curve which is homotopic to a non-simple closed geodesic. We may use the two-fold orientable covering surface to prove that a nontrivial simple closed curve on a non-orientable compact hyperbolic surface is homotopic to either a simple closed geodesic or a 2-fold iterate of a simple closed so-called one-sided geodesic.

Since c is simple, the unique lifting property implies that for some constant co, R(C(T)) = C{T + co), a contradiction. 7. (iv). Since c is not smooth, y n c = 0 , by (ii). By (iii), y is simple. The proof that y and c bound an embedded annulus is of purely topological nature and is postponed to the Appendix (Proposition A. 11). 6 no longer hold. 8 Example. (Cusps). Let S be a non-compact hyperbolic surface and let W c S be a domain which is isometric to the surface (1) ]-oo, 0 ] x 5 ' = ]-oo, 0] x R/[f i-> f + 1] with the Riemannian metric (2) ds2 = dp2 + e2p dt2 (cf.

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