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Summary  

We show that an isometry between not locally hyperkhler, locally irreducible Khler manifolds is either holomorphic or antiholomorphic.

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We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:ℝn→ℝn(2≦n<∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :ℝn→ℝnup to translation such that f(x) = (b/a) I(x).

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Abstract  

It is shown that the group of isometries of the 3-dimensional space with respect to taxicab metric is the semi-direct product of octahedral group O h and T(3), where O h is the (Euclidean) symmetry group of the regular octahedron and T(3) is the group of all translations of the 3-dimensional space.

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The Pons Asinorum , or the Bridge of Asses , refers to Proposition 5 of Book I of Euclid’s Elements . This proposition and its converse, Proposition 6, state that two sides of a triangle are equal if and only if the opposite angles are equal. Analogues of these propositions for higher dimensional d -simplices are considered in this paper, and satisfactory results are obtained for orthocentric d -simplices. These results do not hold for non-orthocentric d -simplices, thus supporting the point of view that orthocentric d -simplices and not arbitrary ones are the adequate generalization of triangles.

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