Riemannian embeddings in codimension one as unbounded $KK$-cycles

Given a codimension one Riemannian embedding of Riemannian spin$^c$-manifolds $\imath:X \to Y$ we construct a family $\{\imath_!^ ε\}_{0< ε< ε_0}$ of unbounded $KK$-cycles from $C(X)$ to $C_0(Y)$, each equipped with a connection $\nabla^ε$ and each representing the shriek class $\imath_! \in KK(C(X), C_0(Y))$. We compute the unbounded product of $\imath_!^ε$ with the Dirac operator $D_Y$ on $Y$ and show that this represents the $KK$-theoretic factorization of the fundamental class $[X] = \imath_! \otimes [Y]$ for all $ε$. In the limit $ε\to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}ε T + D_X + \mathcal{O}(ε)$ where the ``divergent'' part $T$ is an index cycle representing the unit in $KK(\mathbb{C}, \mathbb{C})$ and the constant ``renormalized'' term is the Dirac operator $D_X$ on $X$. The curvature of $(\imath_!^ε, \nabla^ε)$ is further shown to converge to the square of the mean curvature of $\imath$ as $ε\to 0$.

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