By Yu.D. Burago, V.A. Zalgaller

The unique model of this text used to be written greater than fiveyears in the past with S. Z. Shefel',a profound and unique mathematician who died in 1984. Sincethen the geometry of surfaces has persisted to be enriched with rules and effects. This has required alterations and additions, yet has no longer prompted the nature of the item, the layout ofwhich originated with Shefel'. with no understanding to what volume Shefel' might have licensed the adjustments, I may still however prefer to devote this text to his reminiscence. (Yu. D. Burago) we're attempting to kingdom the qualitative questions of the idea of surfaces in Euclidean areas within the shape during which they seem to the authors at the moment. This description doesn't totally correspond to the old improvement of the topic. the idea of surfaces was once constructed within the first position frequently because the three thought of surfaces in 3-dimensional Euclidean house E ; in spite of the fact that, it is sensible to start by way of contemplating surfaces F in Euclidean areas of any measurement n~ three. This strategy permits us, particularly, to install a brand new gentle a few three unsolved difficulties of this built (and when it comes to surfaces in E really whole) concept, and in lots of circumstances to consult the connections with the current degree ofdevelopment of the idea of multidimensional submanifolds. The prime query of the item is the matter of the relationship among periods of metrics and periods of surfaces in En.

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That is, X = L2 (ZΓ\G, ω) is the completion of a direct sum of irreducibles. To see that a π-isotype is of finite multiplicity, again use compactness of the operators from Cco (G). With ϕ = ϕ∗ , R(ϕ) acts by scalar λ on X π (λ). Non-zero scalar operators are compact only for finite-dimensional spaces, so X π (λ) is finite-dimensional for λ = 0. Given π, take ϕ = ϕ∗ such that R(ϕ) is non-zero on the representation space Vπ of π, invoking the existence of approximate identities. Then Vπ (λ) = 0 for some non-zero λ.

Let π be unitary. Then 0≤| ci π(gi )v|2 = ij i ci cj π(gj−1 gi )vi , v ci cj π(gj )∗ π(gi )vi , v = ci cj π(gi )vi , π(gj )v = ij ij by unitariness. On the other hand, suppose f is positive definite. Let ci f (ggi ) E(g) = dj f (ghj ) F (g) = j i Using f (g −1 ) = f (g), ci dj f (h−1 j gi ) = E, F = dj E(h−1 j )= j ij ci F (gi−1 ) i Thus, the inner product E, F is independent of the expressions for E and F , but only depends on the functions. That is, the inner product is well-defined. That the inner product has the positive semi-definiteness property F, F ≥ 0 is immediate from the definition of positive-definiteness.

Traces, characters, central functions for Z ÒG compact Take Z\G compact with Z a closed subgroup of the center of G (so G is unimodular). From above, every irreducible unitary of G is finite dimensional, with some unitary central ω on Z. 1] Definition: Let π, V be an irreducible (finite-dimensional) of G. The character χπ of π is the function on G defined by χπ (g) = trace π(g) = π(g)ei , ei i for any orthonormal basis ei of π. 2] Remark: If π were not finite-dimensional, the character χπ of π could not be defined as a pointwise-evaluatable function, but only as some more general sort of entity.