<p>We study the semi-simple (or non-parabolic) isometries <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> of the manifold <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {P}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of symmetric positive definite real matrices, endowed with the trace metric <i>g</i> and pay attention to the set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Min(\Phi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> consisting of the points minimizing the displacement function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$P \mapsto d(P, \Phi (P))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>↦</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, for every isometry <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\mathcal {P}_n, g)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">P</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we give the explicit expression and study the differential-geometric structure of the set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Min(\Phi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Semi-simple isometries of the manifold of symmetric positive definite real matrices with the trace metric

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2026年6月3日

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Donato Pertici,
Alberto Dolcetti

摘要

We study the semi-simple (or non-parabolic) isometries $\Phi $ $Φ$ of the manifold $\mathcal {P}_n$ $P_{n}$ of symmetric positive definite real matrices, endowed with the trace metric g and pay attention to the set $Min(\Phi )$ $M i n (Φ)$ consisting of the points minimizing the displacement function $P \mapsto d(P, \Phi (P))$ $P \mapsto d (P, Φ (P))$ . In particular, for every isometry $\Phi $ $Φ$ of $(\mathcal {P}_n, g)$ $(P_{n}, g)$ , we give the explicit expression and study the differential-geometric structure of the set $Min(\Phi )$ $M i n (Φ)$ .