Solitons of the midpoint mapping and affine curvature

For a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$ x = ( x j ) j ∈ Z in $$\mathbb {R}^n$$ R n we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$ ( M ( x ) ) j = x j + x j + 1 / 2 . We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$ R n . These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$ c ˙ ( t ) = B c ( t ) + d for a matrix B and a vector d. For $$n=2$$ n = 2 these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$ k ga = k ga ( B ) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.

Prescribing Centro-Affine Curvature From One Convex Body to Another

We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

Curvature tensor of connection in principal bundle of Cartan's projective connection space

We considered Cartan's projective connection space with structure equations generalizing the structure equations of the projective space and the condition of local projectivity (this condition is an analogue to the equiprojectivity condition in the projective space). The curvature-torsion object of the space is a tensor containing three subtensor: torsion tensor, torsion affine curvature tensor, extended torsion tensor. Cartan's projective connection space is not a space with connection of the principal bundle. The assignment of a connection in the adjoint principal bundle leads to a space with a connection. It is proved that the curvature object of the introduced connection is a tensor.