We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups $\mathrm{GA}\left(2\right)=\mathrm{GL}(2,ℝ)⋉{ℝ}^2$ and $\mathrm{GA}\left(3\right)=\mathrm{GL}(3,ℝ)⋉{ℝ}^3$, 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...

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].