论文标题
riemannian嵌入Codimension One中的嵌入方式无限制$ kk $ -cycles
Riemannian embeddings in codimension one as unbounded $KK$-cycles
论文作者
论文摘要
给定一个riemannian旋转$^ c $ -manifolds $ \ imath：x \ y $的riemannian嵌入，我们构建一个家庭$ \ {\ imath _！ $ \ nabla^ε$，每个代表尖叫类$ \ imath_！ \ in Kk（C（X），C_0（y））$。我们将$ \ imath _！^ε$的无界产品与$ y $上的Dirac Operator $ d_y $一起计算，并表明这表示$ kk $ - 理论分解基本类$ [x] = \ imath_！ \ otimes [y] $ for hast $ε$。在极限$ε\至0 $中，产品运营商承认表格$ \ frac {1}εt + d_x + d_x + \ Mathcal {o}（ε）$的渐近扩展，其中`'divergent'part $ t $是$ kk（\ mathbb {c} c的指示单位的``divergent'part $ t $是一个指数``重新归一化的''术语是$ x $上的狄拉克运算符$ d_x $。 $（\ imath _！^ε，\ nabla^ε）$的曲率进一步显示为$ \ imath $的平均曲率正方形，为$ε\ to 0 $。
Given a codimension one Riemannian embedding of Riemannian spin$^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 $ε\to 0$.
