nLab homothety

Definition

Traditional

In the original sense, a homothety is a function from a Euclidean space to itself, which is a dilation centered at a given point of the Euclidean space. Hence a homothety of a Cartesian space $\mathbb{R}^n$ is a function $\mathbb{R}^n \to \mathbb{R}^n$ given by multiplication with some $\kappa \in \mathbb{R}-\{0\}$.

Such dilations are linear functions $\kappa \in GL(\mathbb{R}^n)$ which commute, in particular, with all orthogonal transformations on $\mathbb{R}^n$. In view of this, in the context of Cartan geometry one says more generally that given a vector space $V$ and a group homomorphism $\phi \colon G \longrightarrow GL(V)$ to the general linear group of $V$, then a homothety of this data is an element $\kappa \in GL(V)$ such that for all $g \in G$ one has

$$\kappa \circ \phi(g) = \phi(g) \circ \kappa \,.$$

Given then two G-structures on a smooth manifold modeled on $V$, then a homothety between these $G$-structures is a homothety of the local models which transforms the two $G$-structures into each other.

General abstract

This has a succinct diagrammatic formulation in the formulation of $G$-structures on $V$-manifolds in differential cohesion (see at differential cohesion – G-Structures). Write

$$G \mathbf{Struc} \colon \mathbf{B}G\longrightarrow \mathbf{B}GL(V)$$

for the delooping of the group homomorphism defining the kind of $G$-structure. Then a homothety is simply a homotopy from $G\mathbf{Struc}$ to itself

$$\array{ & & \mathbf{B}G \\ & \swarrow && \searrow^{\mathrlap{G\mathbf{Struc}}} \\ & {}_{\mathllap{G\mathbf{Struc}}}\searrow &\swArrow_{\simeq}& \swarrow \\ && \mathbf{B}GL(V) } \,.$$

Accordingly, for $\mathbf{g}_1, \mathbf{g}_2 \colon \tau_X\longrightarrow G \mathbf{Struc}$ two $G$-structures on a $V$-manifold

$$\array{ X && \longrightarrow && \mathbf{B}G \\ & {}_{\mathllap{\tau_X}}\searrow &\swArrow_{\mathrlap{\mathbf{g}_i}}& \swarrow_{\mathrlap{G}\mathbf{Struc}} \\ && \mathbf{B}GL(V) }$$

then a homothety from $\mathbf{g}_1$ to $\mathbf{g}_2$ is a homotopy of homotopies of the form

$$\array{ X && \longrightarrow && \mathbf{B}G \\ & {}_{\mathllap{\tau_X}} \searrow &\swArrow_{\mathrlap{\mathbf{g}_1}}& \swarrow_{\mathrlap{G\mathbf{Struc}}} &\swArrow_{\simeq}& \searrow^{\mathrlap{G\mathbf{Struc}}} \\ && \mathbf{B}GL(V) && =& \mathbf{B}GL(V) } \;\;\;\;\;\; \stackrel{\simeq}{\Rightarrow} \;\;\;\;\;\; \array{ X && \longrightarrow && \mathbf{B}G \\ & {}_{\mathllap{\tau_X}}\searrow &\swArrow_{\mathrlap{\mathbf{g}_2}}& \swarrow_{\mathrlap{G\mathbf{Struc}}} \\ && \mathbf{B}GL(V) }$$

References

• Encyclopedia of Mathematics, Homothety