By Alfred Barnard Basset

This quantity is made out of electronic pictures from the Cornell collage Library historic arithmetic Monographs assortment.

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**Extra resources for A treatise on the geometry of surfaces**

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Thus spec(ri\G,g) = spec'(ri\G,g) U spec"(ri\G,g), where spec'(ri\G,g) and spec" (r i \ G , g) are defined as the spectrum of the Laplacian restricted to '3£;, and '3£1', respectively. The multiplicity of an eigenvalue in spec(ri\G,g) is equal to the sum of its multiplicities in spec'(ri\G,g) and spec"(ri\G,g). The subspace '3£; is canonically isomorphic to (fi\G), and since the projection from the nilmanifold (ri\G,g) onto the quotient nilmanifold (fi\G,g) is a Riemannian submersion, we have spec(fi\G,g) = spec'(ri\G,g) for i = 1,2.

The Lie group G admits a 2-parameter family of almost inner, non-inner automorphisms given by To see that these are almost inner, note that conjugation in G is given by Thus iI>s,t(h) = h'h(11')-1 with h' = (x~,x;,Y; ,Y;,O,O) where ifxI=Y2=0 if Xl = 0, Y2 i' 0 if X I i' 0, Y2 = 0 if XI i' 0, Y2 i' O. t = iI>;,tg. 7. The changing geometry. t in the example above for a particular choice of initial metric g. 31 investigate how the geometry changes in general for isospectral deformations of two-step nilmanifolds.

The condition that r l and r2 be representation equivalent discrete subgroups of G actually has implications concerning the length spectrum of (r l \G,g) and (r2\G,g). These implications arise from the two conjugacy conditions (L) and (R) defined below. 3. Let r l and r z be cocompact discrete subgroups of the Lie group G with left-invariant metric g. If for each h in G we have #{ blrl C [hlC} then [Ll- spec(rl \G, g) = #{ blr2 C [hlC} , (L) = [L 1- spec(rz \G,g). Here #{ blr; C [hl c } denotes the number of distinct conjugacy classes in ri contained in the conjugacy class of h in G.