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Two remarks on Bravais manifolds. (Russian) Zbl 0374.10022

Let \(G\) be a finite group of integral-valued unimodular matrices of order \(n\). The set \(B_G\) of all quadratic forms \(f(x)\) for which each element of the group \(G\) is an automorphism is called the Bravais manifold of the group \(G\). The following two theorems are proved:
(1) Let \(G\) be defined as above, \(G\not\subset \{\pm I_n\}\) (the unit matrix of order \(n\)), then \(\dim B_G \le N - (n - 1)\) and this bound is actually attained.
(2) Let \(B_{G\varphi}\) be the Bravais manifold of the group of automorphisms \(G_\varphi\) of a given perfect form \(\varphi\). If \(\varphi\) is not extreme, then \(\dim B_{G\varphi} > 1\).

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11H06 Lattices and convex bodies (number-theoretic aspects)
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