Affine differential geometry

Affine differential geometry, as its name suggests, is a type of differential geometry. 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. In order to understand this article a basic knowledge of connexions would be helpful.

Preliminaries

Here we consider the simplest case, i.e. manifolds of codimension one. Let $M\; subset\; mathbb\{R\}^\{n+1\}$ be an $n$-dimensional manifold, let $mathfrak\{X\}(M)$ denote the module of smooth vector fields over $M$, and let $xi$ be a transverse vector field such that $T\_pmathbb\{R\}^\{n+1\}\; =\; T\_pM\; oplus\; langle\; xi\; angle$ for all $p\; in\; M$. We also have the equation where $S\; :\; mathfrak\{X\}(M)\; o\; mathfrak\{X\}(M)$ is the shape operator and $au\; :\; mathfrak\{X\}(M)\; o\; mathbb\{R\}$ is the transverse connexion form.

Let $D\; :\; mathfrak\{X\}(mathbb\{R\}^\{n+1\})^2\; o\; mathfrak\{X\}(mathbb\{R\}^\{n+1\})$ be the standard covarient derivative, where $(X,Y)\; mapsto\; D\_XY$. For $X,\; Y\; in\; mathfrak\{X\}(M)$ we have the Gaussian decomosition $D\_XY\; =\; abla\_XY\; +\; h(X,Y)xi,$ where $abla\; :\; mathfrak\{X\}(M)^2\; o\; mathfrak\{X\}(M)$ is the induced connexion and $h:mathfrak\{X\}(M)^2\; o\; mathbb\{R\}$ is the induced symmetric fundamental form. Notice that $abla$ and $h$ are determined by a choice of $xi.$We also have the equation $D\_X\; xi\; =\; -SX\; +\; au(X)xi$ where $S\; :\; mathfrak\{X\}(M)\; o\; mathfrak\{X\}(M)$ is the shape operator and $au:mathfrak\{X\}(M)\; o\; mathbb\{R\}$ is the transverse connexion form.

The first Induced volume form

Let $Omega\; :\; mathfrak\{X\}(mathbb\{R\}^\{n+1\})^\{n+1\}\; o\; mathbb\{R\}$ be a volume form, and let $X\_i\; in\; mathfrak\{X\}(M)$ for $1\; le\; i\; le\; n$. Then we induce a volume form on $M$ given by $omega\; :\; mathfrak\{X\}(M)^n\; o\; mathbb\{R\}$ where$omega(X\_1,ldots,X\_n)\; :=\; Omega(X\_1,ldots,X\_n,xi)\; .$This is a natural definition: in Euclidean differential geometry where $xi$ is the Euclidean unit normal then the volume spanned by $\{X\_1,ldots,X\_n\}$ is always equal to $omega(X\_1,ldots,X\_n)$.

The second induced volume form

Let $H\; :=\; (h\_\{i,j\})$ where $h\_\{i,j\}\; :=\; h(X\_i,X\_j).$ We define a second volume form given by $u\; :\; mathfrak\{X\}(M)^n\; o\; mathbb\{R\}$ where $u(X\_1,ldots,X\_n)\; :=\; |det(H)|^\{1/2\}.$ This is a natural definition. Let $M\; =\; mathbb\{R\}^2$ and $h(X\_i,X\_j)\; :=\; X\_i\; cdot\; X\_j$ (Euclidean scaler product), then: where $heta$ is the angle between $X\_1$ and $X\_2$. It follows that $u(X\_1,X\_2)\; =\; ||X\_1||\; cdot\; ||X\_2||\; cdot\; sin\; heta.$ This is of course the area of the parallelagram in the plane with sides $X\_1$ and $X\_2$.

Two natural conditions

We now impose two natural conditions. First, we want the connexion $abla$ and the volume form $omega$ to be compatible. That means that $abla\_Xomega\; =\; 0$ for all $X\; in\; mathfrak\{X\}(M).$ This says that $omega$ is invariant under parallel transport with respect to $abla.$ It has been shown (e.g. K. Nomizu) that $abla\_Xomega\; =\; au(X)omega$ and so $abla\_Xomega\; =\; 0$ for all $X\; in\; mathfrak\{X\}(M)\; iff\; D\_Xxi\; in\; mathfrak\{X\}(M)$ for all $X\; in\; mathfrak\{X\}(M).$ The second condition is that $omega\; equiv\; u.$.

The punch line

It can be shown (e.g. K. Nomizu) that the is, up to sign, a unique choice of transverse vector field such that(1) $ablaomega\; equiv\; 0$ and (2) $omega\; equiv\; u\; .$ This unique transverse vector field is called the affine normal vector field. Affine differential geometry is the study of this transverse vector field, the induced connexion, the fundamental form, the shape operator etc. It is easy to show that these abojects are invariant under special affine transformation, i.e. $mbox\{SL\}(n+1,mathbb\{R\})\; ltimes\; mathbb\{R\}^\{n+1\}.$

References

*Citation|first=K.|last=Nomizu|first2=T.|last2=Sasaki|title=Affine Differential Geometry: Geometry of Affine Immersions |publisher=Cambridge University Press|year=1994|ISBN=0521441773

*Citation|first=V.|last=Ovisienko|first2=S.|last2=Tabachnikov|title=Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups|publisher=Cambridge University Press|year=2004|ISBN=0521831865

*Citation|first=Buchin|last=Su|title=Affine Differential Geometry|publisher=Harwood Academic|year=1983|ISBN=0677310609

See also

* Affine geometry of curves