Poisson transversals are submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In this communication, we prove a normal form theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous normal forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the normal form data of the transversals. Our second result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a normal form theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all structure maps in normal form. We conclude by illustrating our results with examples arising from Lie algebras.

中文翻译：

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Poisson 映射和围绕 Poisson 横断面的辛群的正规形式

泊松横向是泊松流形中的子流形，它与所有辛叶横向和辛相交。在这次交流中，我们证明了泊松横截面周围泊松映射的范式定理。泊松图将泊松横截面拉回泊松横截面，我们的第一个主要结果表明，在这样的横截面周围存在同时存在的正常形式，为此泊松图变为横向线性，并且交织了横截面的正常形式数据。我们的第二个结果涉及辛积分。我们证明泊松横截面的邻域是可积的，当泊松横截面本身是可积的，在这种情况下，我们证明了辛群形围绕其对泊松横截面的限制的范式定理，它将所有结构映射置于正常形式。最后，我们用李代数的例子来说明我们的结果。