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Projected connection and Kato transport are well defined

Statement

Let $M$ be a finite-dimensional smooth manifold, let $d \geq 1$ , and let $P : M \to End (C^{d})$ be a $C^{2}$ map satisfying $P (x)^{2} = P (x) = P (x)^{*}$ and $rank P (x) = r$ for a fixed $r$ . Set $E_{P} = {(x, v) \in M \times C^{d} : P (x) v = v}, π (x, v) = x .$ Here a rank- $r$ $C^{2}$ complex vector bundle means a surjection that has local trivializations over open subsets of $M$ with fiber maps complex-linear and transition matrices of class $C^{2}$ . Then $E_{P} \to M$ is such a bundle. On each open $U \subseteq M$ , the projected operation $\nabla s = P d s$ is complex-linear from $C^{1}$ sections to continuous $E_{P}$ -valued one-forms, satisfies $\nabla (f s) = d f s + f \nabla s$ for $f \in C^{1} (U, C)$ , and satisfies $d (s^{*} t) = (\nabla s)^{*} t + s^{*} \nabla t$ for $C^{1}$ sections $s, t$ ; these are the connection and Hermitian compatibility conditions. For every $C^{2}$ path $γ : [0, 1] \to M$ whose derivatives extend continuously to the endpoints, the Kato equation $W^{'} (t) = [P_{γ}^{'} (t), P_{γ} (t)] W (t), W (0) = I_{d},$ has a unique $C^{2}$ solution. That solution is unitary and satisfies $P_{γ} (t) W (t) = W (t) P_{γ} (0)$ for every $t \in [0, 1]$ .

Statement references

C2 projector bundles and Kato transport

Facts & Assumptions

Given: A finite-dimensional smooth manifold $M$ , an integer $d \geq 1$ , a $C^{2}$ constant-rank orthogonal projection family $P$ , and a $C^{2}$ path $γ : [0, 1] \to M$ .

[F1]

C2 projector bundles and Kato transport: $P^{2} = P = P^{*}$ , its rank is constant, $E_{P, x} = ran P (x)$ , $\nabla s = P d s$ , and the Kato equation uses $K = [P_{γ}^{'}, P_{γ}]$ .

Proof

1.1givenF1constructalgebra

Fix $x_{0} \in M$ and choose an orthonormal basis $e_{1}, \dots, e_{r}$ of $ran P (x_{0})$ . The vectors $P (x) e_{j}$ depend $C^{2}$ on $x$ . Their Gram matrix is the identity at $x_{0}$ , so its determinant remains positive on a neighborhood of $x_{0}$ ; hence the vectors remain linearly independent there. Applying the finite Gram–Schmidt formulas uses only sums, products, division by positive norms, and square roots of positive functions, so it gives a $C^{2}$ orthonormal frame $V (x) = (v_{1} (x), \dots, v_{r} (x))$ for $ran P (x)$ on that neighborhood. The local bundle chart is $Φ (x, z) = (x, V (x) z)$ , with inverse $Φ^{- 1} (x, v) = (x, V (x)^{*} v)$ ; both maps are $C^{2}$ . On an overlap, the frame change is $z \mapsto V_{β} (x)^{*} V_{α} (x) z$ , a $C^{2}$ map into the invertible complex matrices, so the transition maps are $C^{2}$ . These charts make $E_{P}$ a rank- $r$ $C^{2}$ complex vector bundle. For $r = 0$ the bundle is the zero bundle, and for $r = d$ every $P (x) = I_{d}$ and the bundle is trivial.

1.2givenF1constructalgebra

Along $γ$ , abbreviate $P_{γ}$ by $P$ . Differentiating $P^{2} = P$ gives $P^{'} P + P P^{'} = P^{'}$ and $P P^{'} P = 0$ . Therefore $K = P^{'} P - P P^{'}$ is $C^{1}$ , $K^{*} = - K$ , and $[K, P] = P^{'} P + P P^{'} = P^{'}$ . Let $M_{K} = \sup_{t \in [0, 1]} ∥ K (t) ∥$ , set $V_{0} (t) = I_{d}$ , and recursively set $V_{n + 1} (t) = \int_{0}^{t} K (u) V_{n} (u) d u$ . Induction gives $∥ V_{n} (t) ∥ \leq (M_{K} t)^{n} / n!$ , so $W = \sum_{n = 0}^{\infty} V_{n}$ converges uniformly and satisfies $W (t) = I_{d} + \int_{0}^{t} K (u) W (u) d u$ . The fundamental theorem of calculus gives $W^{'} = K W$ ; since $K \in C^{1}$ , also $W \in C^{2}$ . If $Z$ is the difference of two solutions, iteration of $Z (t) = \int_{0}^{t} K (u) Z (u) d u$ gives $∥ Z (t) ∥ \leq \sup_{[0, 1]} ∥ Z ∥ (M_{K} t)^{n} / n!$ for every $n$ , whose right side tends to zero, so $Z = 0$ .

2.1F1step 1.1algebra

For a $C^{1}$ scalar function $f$ and a $C^{1}$ section $s$ with $P s = s$ , the product rule gives $\nabla (f s) = P d (f s) = P (d f s + f d s) = d f s + f \nabla s$ , so $\nabla$ satisfies the connection Leibniz rule. For two such sections $s, t$ , $d (s^{*} t) = (P d s)^{*} t + s^{*} (P d t)$ because $P = P^{*}$ and $P s = s$ , $P t = t$ ; thus $\nabla$ preserves the Hermitian fiber metric.

3.1F1step 1.2algebra∎

Differentiation yields $(W^{*} W)^{'} = W^{*} (K^{*} + K) W = 0$ , hence $W^{*} W = I_{d}$ ; a square matrix with a left inverse is invertible, so $W$ is unitary. Also $(W^{*} P W)^{'} = W^{*} (P^{'} - [K, P]) W = 0$ , so $W (t)^{*} P (t) W (t) = P (0)$ . Multiplying by $W (t)$ on the left gives $P (t) W (t) = W (t) P (0)$ , as required.

Depends on

C2 projector bundles and Kato transport