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General-affine invariants of plane curves and space curves

Shimpei Kobayashi, Takeshi Sasaki (2020)

Czechoslovak Mathematical Journal

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups GA ( 2 ) = GL ( 2 , ) 2 and GA ( 3 ) = GL ( 3 , ) 3 , respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective...

Grassmann manifold V 3 4 in the projective space P 7 with characteristics consisting of a quadric and two planes

Josef Vala (1993)

Mathematica Bohemica

Some results in the geometry of four-parametric manifolds of three-dimensional spaces in the projective space P 7 are found. The properties of such a manifold V 3 4 with characteristics consisting of a quadric and two planes are studied. The properties of the manifold dual to V 3 4 are found. Some results in the geometry of linear spaces from [1],[2],[3],[4] are used. The notation of the quantities is the same as in [4].

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