Affine Mappings. Submanifolds

Abstract

Let χ and y be affine connection spaces with the connections ∇^χ and ∇^y (to simplify formulas, we often write ∇ instead of ∇^χ and $\hat \nabla$ instead of ∇^y). On each coordinate neighborhood U of the manifold χ (coordinate neighborhood V of the manifold y), the connection ∇^χ (connection ∇^y) is given by the matrix ω = ω ^x (matrix $\hat \omega$= ω ^y) of connection forms. The connection ∇^x sets the horizontal subspace H ^x _A of the tangent space T _A (T χ) in correspondence with each tangent vector A (point of the total space T χ of the tangent bundle _τx). Similarly, the connection ∇^y sets the horizontal subspace H ^y _B ⊂ T _B (T y) in correspondence with each point B ∈ T y.