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<p>Let <i>G</i><sub><i>q</i></sub>(ℝ<sup><i>n</i></sup>) be the Grassmannian of all linear <i>q</i> dimensional subspaces of ℝ<sup><i>n</i></sup> and <i>I</i> an integral invariant of <i>p</i> + <i>q</i> − <i>n</i> dimensional submanifolds of ℝ<sup><i>n</i></sup>. Then we give methods of evaluating Crofton type integral</p><p><Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_112_Article_Equa.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </MediaObject> <EquationSource Format="TEX">\(\int_{{G_q}\left({{\mathbb{R}^n}} \right)} {I\left({M \cap L} \right){\rm{d}}L.}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mo>∫</mo> <mrow> <mrow> <msub> <mi>G</mi> <mi>q</mi> </msub> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mo>∩</mo> <mi>L</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mrow> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>L</mi> <mo>.</mo> </mrow> </math></EquationSource> </Equation></p><p>The methods also work for various generalizations of <i>G</i><sub><i>q</i></sub>(ℝ<sup><i>n</i></sup>) such as complex Grassmannians.</p>

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Kinematic and Crofton formulas for linear groups

  • 期刊论文

  • 发表日期：

    2025年1月24日

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  • Ralph Howard

摘要

Let Gq(ℝn) be the Grassmannian of all linear q dimensional subspaces of ℝn and I an integral invariant of p + q − n dimensional submanifolds of ℝn. Then we give methods of evaluating Crofton type integral

\(\int_{{G_q}\left({{\mathbb{R}^n}} \right)} {I\left({M \cap L} \right){\rm{d}}L.}\) ∫ G q ( R n ) I ( M ∩ L ) d L .

The methods also work for various generalizations of Gq(ℝn) such as complex Grassmannians.