nLab
homothety

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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 n\mathbb{R}^n is a function n n\mathbb{R}^n \to \mathbb{R}^n given by multiplication with some κ{0}\kappa \in \mathbb{R}-\{0\}.

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

κϕ(g)=ϕ(g)κ. \kappa \circ \phi(g) = \phi(g) \circ \kappa \,.

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

General abstract

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

GStruc:BGBGL(V) G \mathbf{Struc} \colon \mathbf{B}G\longrightarrow \mathbf{B}GL(V)

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

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

Accordingly, for g 1,g 2:τ XGStruc\mathbf{g}_1, \mathbf{g}_2 \colon \tau_X\longrightarrow G \mathbf{Struc} two GG-structures on a VV-manifold

X BG τ X g i GStruc BGL(V) \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 g 1\mathbf{g}_1 to g 2\mathbf{g}_2 is a homotopy of homotopies of the form

X BG τ X g 1 GStruc GStruc BGL(V) = BGL(V)X BG τ X g 2 GStruc BGL(V) \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

Last revised on May 30, 2015 at 02:41:51. See the history of this page for a list of all contributions to it.