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Affine differential geometry, is a type of differential geometry in which the differential invariants are invariant under volume-preserving affine transformations. The name ''affine differential geometry'' follows from Klein's Erlangen program. The basic difference between affine and Riemannian differential geometry is that in the affine case we introduce volume forms over a manifold instead of metrics. ==Preliminaries== Here we consider the simplest case, i.e. manifolds of codimension one. Let be an ''n''-dimensional manifold, and let ξ be a vector field on transverse to such that for all where ⊕ denotes the direct sum and Span the linear span. For a smooth manifold, say ''N'', let Ψ(''N'') denote the module of smooth vector fields over ''N''. Let be the standard covariant derivative on R''n''+1 where We can decompose ''DXY'' into a component tangent to ''M'' and a transverse component, parallel to ξ. This gives the equation of Gauss: where is the induced connexion on ''M'' and is a bilinear form. Notice that ∇ and ''h'' depend upon the choice of transverse vector field ξ. We consider only those hypersurfaces for which ''h'' is non-degenerate. Interestingly, this is a property of the hypersurface ''M'' and does not depend upon the choice of transverse vector field ξ. If ''h'' is non-degenerate then we say that ''M'' is non-degenerate. In the case of curves in the plane, the non-degenerate curves are those without inflexions. In the case of surfaces in 3-space, the non-degenerate surfaces are those without parabolic points. We may also consider the derivative of ξ in some tangent direction, say ''X''. This quantity, ''D''''X''ξ, can be decomposed into a component tangent to ''M'' and a transverse component, parallel to ξ. This gives the Weingarten equation: The type-(1,1)-tensor is called the affine shape operator, the differential one-form is called the transverse connexion form. Again, both ''S'' and τ depend upon the choice of transverse vector field ξ. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Affine differential geometry」の詳細全文を読む スポンサード リンク
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