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

TL;DR

This paper constructs a family of unbounded KK-cycles for codimension one Riemannian embeddings, analyzes their product with Dirac operators, and studies their asymptotic behavior and curvature convergence as a parameter approaches zero.

Contribution

It introduces a novel family of unbounded KK-cycles for Riemannian embeddings and analyzes their product with Dirac operators, revealing asymptotic expansion and curvature convergence.

Findings

Asymptotic expansion of the product operator as epsilon approaches zero.

Convergence of curvature to the mean curvature squared.

Representation of the fundamental class via KK-theoretic factorization.

Abstract

Given a codimension one Riemannian embedding of Riemannian spin$^c$-manifolds $\imath:X \to Y$ we construct a family $\{\imath_!^ \epsilon\}_{0< \epsilon< \epsilon_0}$ of unbounded $KK$-cycles from $C(X)$ to $C_0(Y)$, each equipped with a connection $\nabla^\epsilon$ and each representing the shriek class $\imath_! \in KK(C(X), C_0(Y))$. We compute the unbounded product of $\imath_!^\epsilon$ 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 $\epsilon$. In the limit $\epsilon \to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}{\epsilon} T + D_X + \mathcal{O}(\epsilon)$ 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 $(\imath_!^\epsilon, \nabla^\epsilon)$ is further shown to converge to the square of the mean curvature of $\imath$ as $\epsilon \to 0$.