C2 projector bundles and Kato transport

Let $M$ be a finite-dimensional smooth manifold, let $d\ge 1$ be an integer, and give ${\mathbb{C}}^d$ its standard Hermitian inner product. Let $P:M\to \mathrm{End}({\mathbb{C}}^d)$ be a $C^2$ map such that $P(x{)}^2=P(x)=P(x{)}^{\ast }$ and $\mathrm{rank}P(x)=r$ is constant. Define $E_P=\{(x,v)\in M\times {\mathbb{C}}^d:P(x)v=v\},\pi (x,v)=x\mathrm{.}$ Its fiber at $x$ is $\mathrm{ran}P(x)$ with the inner product inherited from ${\mathbb{C}}^d$, using $⟨v,w⟩=v^{\ast }w$. The following finite-regularity convention fixes the meaning of $d$ here. If $s:U\to {\mathbb{C}}^d$ is $C^1$, then $d{s}_x:T_{x}M\to {\mathbb{C}}^d$ is its differential, whose coordinate coefficients are continuous. If a $C^1$ one-form is written in coordinates as $\alpha ={\sum }_{j=1}^n{\alpha }_{j}d{x}^j$, define $d\alpha ={\sum }_{1\le j<k\le n}({\mathrm{\partial }}_j{\alpha }_k-{\mathrm{\partial }}_k{\alpha }_j)d{x}^j\wedge d{x}^k$ componentwise, whenever its coefficients are $C^1$. These derivatives have continuous coefficients. The chain rule and symmetry of mixed second derivatives make the definitions independent of $C^2$ coordinate changes; on smooth forms they agree with the componentwise exterior derivative. Wedge products of these forms are the pointwise alternating products, with matrix multiplication in the coefficient factors when present. In particular, $dP$ is a $C^1$ matrix-valued one-form.

For an open $U\subseteq M$, a $C^1$ section of $E_P$ is a $C^1$ map $s:U\to {\mathbb{C}}^d$ with $Ps=s$. A Hermitian connection at this regularity is a complex-linear operation from such sections to continuous $E_P$-valued one-forms satisfying $\mathrm{\nabla }(fs)=dfs+f\mathrm{\nabla }s$ for $f\in C^1(U,\mathbb{C})$ and $d(s^{\ast }t)=(\mathrm{\nabla }s{)}^{\ast }t+s^{\ast }\mathrm{\nabla }t$ for all $C^1$ sections $s,t$. Define the projected connection by $\mathrm{\nabla }s=Pds$.

For a $C^2$ path $\gamma :[0,1]\to M$, with derivatives extending continuously to the endpoints, put $P_{\gamma }(t)=P(\gamma (t))$ and $K_{\gamma }(t)=[P_{\gamma }^{\mathrm{\prime }}(t),P_{\gamma }(t)]$. Kato transport along $\gamma$ is the matrix solution $W_{\gamma }^{\mathrm{\prime }}(t)=K_{\gamma }(t)W_{\gamma }(t)$, $W_{\gamma }(0)=I_d$.

The connection and Kato transport are defined from the projector family itself; no Schrödinger Hamiltonian or physical time evolution is needed. For a one-parameter $C^2$ family $H:[0,1]\to \mathrm{Herm}({\mathbb{C}}^d)$, with derivatives through order two continuous up to the endpoints, suppose $P(s)$ is the orthogonal spectral projection onto a subset of the spectrum, is continuous in operator norm, and that subset has distance at least $g>0$ from its complementary spectrum for every $s$. Under Countable Choice, the earlier finite-dimensional adiabatic result gives $P\in C^2([0,1],\mathrm{End}({\mathbb{C}}^d))$ of constant rank; restriction to $M=(0,1)$ supplies an example of the projector families defined here.

Smooth manifolds and their smooth charts

$C^r$ and smooth maps between smooth manifolds

Finite-dimensional vector-valued forms and their exterior derivative

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Finite gapped adiabatic estimate