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In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice versa.
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In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties.
In mathematics, the mathematician Sophus Lie initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan.
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In algebraic geometry, Sumihiro's theorem, introduced by, states that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets.
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References
- Luna, Domingo (1973), "Slices étales", Sur les groupes algébriques, Bull. Soc. Math. France, Paris, Mémoire, vol. 33, Paris: Société Mathématique de France, pp. 81–105
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