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# A treatise on the geometry of surfaces by Alfred Barnard Basset

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.