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<p>We define a Chern–Simons invariant of connections on stably trivial vector bundles over smooth manifolds, taking values in 3-forms modulo closed forms with integral cohomology class. We show an additivity property of this invariant for connections defined on a direct sum of bundles, under a certain block-diagonality condition on the curvature. As a corollary, we deduce an obstruction for conformally immersing a <i>n</i>-dimensional Riemannian manifold in a translation manifold of dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Invariants of stably trivial vector bundles with connection

  • 期刊论文

  • 发表日期：

    2026年4月8日

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  • Sergiu Moroianu

摘要

We define a Chern–Simons invariant of connections on stably trivial vector bundles over smooth manifolds, taking values in 3-forms modulo closed forms with integral cohomology class. We show an additivity property of this invariant for connections defined on a direct sum of bundles, under a certain block-diagonality condition on the curvature. As a corollary, we deduce an obstruction for conformally immersing a n-dimensional Riemannian manifold in a translation manifold of dimension \(n+1\) n + 1 .