A generalization of manifolds with corners
<p>In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\...
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| स्वरूप: | Journal article |
| प्रकाशित: |
Elsevier
2016
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| _version_ | 1826290320452616192 |
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| author | Joyce, D |
| author_facet | Joyce, D |
| author_sort | Joyce, D |
| collection | OXFORD |
| description | <p>In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. Manifolds with corners $X$ have boundaries $\partial X$, also manifolds with corners, with $\mathop{\rm dim}\partial X=\mathop{\rm dim} X-1$.</p> <p>We introduce a new notion of &apos;manifolds with generalized corners&apos;, or &apos;manifolds with g-corners&apos;, extending manifolds with corners, which form a category $\bf Man^{gc}$ with ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}\subset{\bf Man^{gc}}$. Manifolds with g-corners are locally modelled on $X_P=\mathop{\rm Hom}_{\bf Mon}(P,[0,\infty))$ for $P$ a weakly toric monoid, where $X_P\cong[0,\infty)^k\times{\mathbb R}^{n-k}$ for $P={\mathbb N}^k\times{\mathbb Z}^{n-k}$.</p> <p>Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries $\partial X$. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in $\bf Man^{gc}$ exist under much weaker conditions than in $\bf Man^c$.</p> <p>This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of $J$-holomorphic curves can be manifolds or Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than ordinary corners.</p> <p>Our manifolds with g-corners are related to the &apos;interior binomial varieties &apos; of Kottke and Melrose in arXiv:1107.3320.</p> |
| first_indexed | 2024-03-07T02:42:23Z |
| format | Journal article |
| id | oxford-uuid:aae43930-40d8-4e31-a00c-6b45b5b36ed8 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:42:23Z |
| publishDate | 2016 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:aae43930-40d8-4e31-a00c-6b45b5b36ed82022-03-27T03:18:14ZA generalization of manifolds with cornersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:aae43930-40d8-4e31-a00c-6b45b5b36ed8Symplectic Elements at OxfordElsevier2016Joyce, D<p>In conventional Differential Geometry one studies manifolds, locally modelled on ${\mathbb R}^n$, manifolds with boundary, locally modelled on $[0,\infty)\times{\mathbb R}^{n-1}$, and manifolds with corners, locally modelled on $[0,\infty)^k\times{\mathbb R}^{n-k}$. They form categories ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}$. Manifolds with corners $X$ have boundaries $\partial X$, also manifolds with corners, with $\mathop{\rm dim}\partial X=\mathop{\rm dim} X-1$.</p> <p>We introduce a new notion of &apos;manifolds with generalized corners&apos;, or &apos;manifolds with g-corners&apos;, extending manifolds with corners, which form a category $\bf Man^{gc}$ with ${\bf Man}\subset{\bf Man^b}\subset{\bf Man^c}\subset{\bf Man^{gc}}$. Manifolds with g-corners are locally modelled on $X_P=\mathop{\rm Hom}_{\bf Mon}(P,[0,\infty))$ for $P$ a weakly toric monoid, where $X_P\cong[0,\infty)^k\times{\mathbb R}^{n-k}$ for $P={\mathbb N}^k\times{\mathbb Z}^{n-k}$.</p> <p>Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries $\partial X$. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in $\bf Man^{gc}$ exist under much weaker conditions than in $\bf Man^c$.</p> <p>This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of $J$-holomorphic curves can be manifolds or Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than ordinary corners.</p> <p>Our manifolds with g-corners are related to the &apos;interior binomial varieties &apos; of Kottke and Melrose in arXiv:1107.3320.</p> |
| spellingShingle | Joyce, D A generalization of manifolds with corners |
| title | A generalization of manifolds with corners |
| title_full | A generalization of manifolds with corners |
| title_fullStr | A generalization of manifolds with corners |
| title_full_unstemmed | A generalization of manifolds with corners |
| title_short | A generalization of manifolds with corners |
| title_sort | generalization of manifolds with corners |
| work_keys_str_mv | AT joyced ageneralizationofmanifoldswithcorners AT joyced generalizationofmanifoldswithcorners |