We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in KK-theory. This construction works in general for symmetric operators (i.e. without assuming self-adjointness), and extends known results for submersions with compact fibers. The assumption of regularity for the vertically elliptic operator is not always satisfied, but depends on the topology and geometry of the submersion, and we give explicit examples of non-regular operators. We apply our main result to obtain a factorization in unbounded KK-theory of the fundamental class of a Riemannian submersion, as a Kasparov product of the shriek map of the submersion and the fundamental class of the base manifold.

中文翻译：

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卡斯帕罗夫产品关于开放歧管的浸没

我们研究了（可能是非紧的和不完整的）黎曼流形上的 Kasparov 乘积。具体来说，我们在黎曼流形的浸没上表明，总空间上的规则垂直椭圆算子和基空间上的椭圆算子的张量和表示相应类的 Kasparov 乘积KK-理论。这种结构通常适用于对称算子（即不假设自伴随性），并扩展了具有紧凑纤维的浸没的已知结果。垂直椭圆算子的规律性假设并不总是得到满足，而是取决于浸没的拓扑和几何形状，我们给出了非常规算子的明确示例。我们应用我们的主要结果来获得无界的分解KK- 黎曼浸没的基本类理论，作为浸没的尖叫图和基本流形的基本类的卡斯帕罗夫乘积。