Mathematische Nachrichten最新文献

{"title":"Calderón–Zygmund theory on some Lie groups of exponential growth","authors":"Filippo De Mari,&nbsp;Matteo Levi,&nbsp;Matteo Monti,&nbsp;Maria Vallarino","doi":"10.1002/mana.202300499","DOIUrl":"https://doi.org/10.1002/mana.202300499","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>=</mo>\u0000 <mi>N</mi>\u0000 <mo>⋊</mo>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 <annotation>$G = N rtimes A$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> is a stratified Lie group and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>R</mi>\u0000 <mo>+</mo>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$A= mathbb {R}_+$</annotation>\u0000 </semantics></math> acts on <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> via automorphic dilations. We prove that the group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> has the Calderón–Zygmund property, in the sense of Hebisch and Steger, with respect to a family of flow measures and metrics. This generalizes in various directions previous works by Hebisch and Steger and Martini et al., and provides a new approach in the development of the Calderón–Zygmund theory in Lie groups of exponential growth. We also prove a weak-type (1,1) estimate for the Hardy–Littlewood maximal operator naturally arising in this setting.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 1","pages":"113-141"},"PeriodicalIF":0.8,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300499","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143111990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Disjoint \u0000 \u0000 p\u0000 $p$\u0000 -convergent operators and their adjoints","authors":"Geraldo Botelho,&nbsp;Luis Alberto Garcia,&nbsp;Vinícius C. C. Miranda","doi":"10.1002/mana.202300561","DOIUrl":"https://doi.org/10.1002/mana.202300561","url":null,"abstract":"&lt;p&gt;First, we give conditions on a Banach lattice &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;annotation&gt;$E$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; so that an operator &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;annotation&gt;$T$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; from &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;annotation&gt;$E$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; to any Banach space is disjoint &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convergent if and only if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;annotation&gt;$T$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is almost Dunford–Pettis. Then, we study when adjoints of positive operators between Banach lattices are disjoint &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;annotation&gt;$E$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;annotation&gt;$F$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;: (i) a positive operator &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$T: E rightarrow F$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is almost weak &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convergent if and only if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$T^*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is disjoint &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-convergent; (ii) &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$E^*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has order continuous norm or &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$F^*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has the positive Schur property of order &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4766-4777"},"PeriodicalIF":0.8,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142862177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence and regularity of strict solutions for a class of fractional evolution equations","authors":"Guang Meng Wu,&nbsp;Jia Wei He","doi":"10.1002/mana.202400074","DOIUrl":"https://doi.org/10.1002/mana.202400074","url":null,"abstract":"<p>We study the existence and Hölder regularity of solutions for fractional evolution equations of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$alpha in (1,2)$</annotation>\u0000 </semantics></math>. By means of an analytic resolvent, we construct an interpolation space, which can effectively lower the regularity of initial data. By virtue of the interpolation space and some properties of the analytic resolvent, we derive the existence and Hölder regularity of strict solutions for an inhomogeneous problem, as well as the existence and Hölder regularity of a nonlinear problem.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4730-4749"},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two-sided estimates of Lebesque constants for Hölder spaces","authors":"Evgenii I. Berezhnoi","doi":"10.1002/mana.202200286","DOIUrl":"https://doi.org/10.1002/mana.202200286","url":null,"abstract":"<p>A two-sided estimate of Lebesgue constants is proposed for the Hölder space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>H</mi>\u0000 <mi>k</mi>\u0000 <mi>ω</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H_k^omega (X)$</annotation>\u0000 </semantics></math>, constructed from the modulus of continuity of the <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>th order calculated in the symmetric space <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math>. Choosing different function spaces <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math>, we obtain new criteria for the convergence of classical Fourier series.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4750-4765"},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical","authors":"Beatrice Brienza,&nbsp;Anna Fino","doi":"10.1002/mana.202400349","DOIUrl":"https://doi.org/10.1002/mana.202400349","url":null,"abstract":"<p>In this paper, we study strong Kähler with torsion (SKT) and generalized Kähler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of <span></span><math>\u0000 <semantics>\u0000 <mi>J</mi>\u0000 <annotation>$J$</annotation>\u0000 </semantics></math>-invariant nilradical and non-<span></span><math>\u0000 <semantics>\u0000 <mi>J</mi>\u0000 <annotation>$J$</annotation>\u0000 </semantics></math>-invariant nilradical. A classification of such SKT Lie algebras in dimension 6 is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized Kähler compact solvmanifolds.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4705-4729"},"PeriodicalIF":0.8,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400349","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the differential geometry of smooth ruled surfaces in 4-space","authors":"Jorge Luiz Deolindo-Silva","doi":"10.1002/mana.202400295","DOIUrl":"https://doi.org/10.1002/mana.202400295","url":null,"abstract":"<p>A smooth ruled surface in 4-space has only parabolic points or inflection points of the real type. We show, by means of contact with transverse planes, that at a parabolic point, there exist two tangent directions determining two planes along which the parallel projection exhibits <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math>-singularities of type butterfly or worse. In particular, such parabolic points can be classified as butterfly hyperbolic, parabolic, or elliptic points depending on the value of the discriminant of a binary differential equation (BDE). Also, whenever such discriminant is positive, we ensure that the integral curves of these directions form a pair of foliations on the ruled surface. Moreover, the set of points that nullify the discriminant is a regular curve transverse to the regular curve formed by inflection points of the real type. Finally, using a particular projective transformation, we obtain a simple parametrization of the ruled surface such that the moduli of its 5-jet identify a butterfly hyperbolic/parabolic/elliptic point, as well as we get the stable configurations of the solutions of BDE in the discriminant curve.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4689-4704"},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A remark on the product by the hyperplane class in the Chow ring of some complete intersections","authors":"René Mboro","doi":"10.1002/mana.202400041","DOIUrl":"https://doi.org/10.1002/mana.202400041","url":null,"abstract":"<p>By classical calculation, for a smooth hypersurface <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Y</mi>\u0000 <mo>⊂</mo>\u0000 <msubsup>\u0000 <mi>P</mi>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$Ysubset mathbb {P}^{n+1}_{mathbb {C}}$</annotation>\u0000 </semantics></math>, the product by the hyperplane class is zero on homologically trivial rational cycles, that is, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>·</mo>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>Y</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>:</mo>\u0000 <msub>\u0000 <mi>CH</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mtext>hom</mtext>\u0000 <mo>,</mo>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>→</mo>\u0000 <msub>\u0000 <mi>CH</mi>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mtext>hom</mtext>\u0000 <mo>,</mo>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$cdot H_{|Y}:{rm CH}_i(Y)_{text{hom},mathbb {Q}}rightarrow {rm CH}_{i-1}(Y)_{text{hom},mathbb {Q}}$</annotation>\u0000 </semantics></math> is 0 for any <span></span><math>\u0000 <semantics>\u0000 <mi>i</mi>\u0000 <annotation>$i$</annotation>\u0000 </semantics></math>. This note extends that result to some complete intersections.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"2014-2036"},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global solvability and hypoellipticity for evolution operators on tori and spheres","authors":"Alexandre Kirilov,&nbsp;André Pedroso Kowacs,&nbsp;Wagner Augusto Almeida de Moraes","doi":"10.1002/mana.202300506","DOIUrl":"https://doi.org/10.1002/mana.202300506","url":null,"abstract":"<p>In this paper, we investigate global properties of a class of evolution differential operators defined on a product of tori and spheres. We present a comprehensive characterization of global solvability and hypoellipticity, providing necessary and sufficient conditions that involve Diophantine conditions and the connectedness of sublevel sets associated with the coefficients of the operator. Furthermore, we recover well-known results from existing literature and introduce novel contributions.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4605-4650"},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Noether–Lefschetz locus of surfaces in \u0000 \u0000 \u0000 P\u0000 3\u0000 \u0000 ${mathbb {P}}^3$\u0000 formed by determinantal surfaces","authors":"Manuel Leal,&nbsp;César Lozano Huerta,&nbsp;Montserrat Vite","doi":"10.1002/mana.202400132","DOIUrl":"https://doi.org/10.1002/mana.202400132","url":null,"abstract":"<p>We compute the dimension of certain components of the family of smooth determinantal degree <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> surfaces in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>${mathbb {P}}^3$</annotation>\u0000 </semantics></math>, and show that each of them is the closure of a component of the Noether–Lefschetz locus <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mi>L</mi>\u0000 <mo>(</mo>\u0000 <mi>d</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$NL(d)$</annotation>\u0000 </semantics></math>. Our computations exhibit that smooth determinantal surfaces in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <annotation>${mathbb {P}}^3$</annotation>\u0000 </semantics></math> of degree 4 form a divisor in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <msub>\u0000 <mi>O</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 </msub>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>4</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$|mathcal {O}_{{mathbb {P}}^3}(4)|$</annotation>\u0000 </semantics></math> with five irreducible components. We will compute the degrees of each of these components: <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>320</mn>\u0000 <mo>,</mo>\u0000 <mn>2508</mn>\u0000 <mo>,</mo>\u0000 <mn>136512</mn>\u0000 <mo>,</mo>\u0000 <mn>38475</mn>\u0000 </mrow>\u0000 <annotation>$320,2508,136512,38475$</annotation>\u0000 </semantics></math>, and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>320112</mn>\u0000 </mrow>\u0000 <annotation>$hskip.001pt 320112$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4671-4688"},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Curvature and Weitzenböck formula for spectral triples","authors":"Bram Mesland,&nbsp;Adam Rennie","doi":"10.1002/mana.202400158","DOIUrl":"https://doi.org/10.1002/mana.202400158","url":null,"abstract":"<p>Using the Levi-Civita connection on the noncommutative differential 1-forms of a spectral triple <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>B</mi>\u0000 <mo>,</mo>\u0000 <mi>H</mi>\u0000 <mo>,</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(mathcal {B},mathcal {H},mathcal {D})$</annotation>\u0000 </semantics></math>, we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac spectral triples and derive a general Weitzenböck formula for them. We apply these tools to <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math>-deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under <span></span><math>\u0000 <semantics>\u0000 <mi>θ</mi>\u0000 <annotation>$theta$</annotation>\u0000 </semantics></math>-deformation, whereas the connection Laplacian, Clifford representation of the curvature, and the scalar curvature are all invariant under deformation.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 12","pages":"4582-4604"},"PeriodicalIF":0.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400158","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142860667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}