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Projector curvature and frame covariance

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 $P^{2} = P = P^{*}$ and $rank P = r$ for a fixed $r$ . Write $E_{P, x} = ran P (x)$ and define $\nabla_{X} s = P d s (X)$ for a $C^{1}$ section $s$ and a vector field $X$ . For a $C^{2}$ section $s$ and smooth vector fields $X, Y$ , define the curvature by $F_{\nabla} (X, Y) s = \nabla_{X} (\nabla_{Y} s) - \nabla_{Y} (\nabla_{X} s) - \nabla_{[X, Y]} s .$ As an endomorphism of $E_{P, x}$ extended by zero on $E_{P, x}^{⊥}$ , it is the continuous two-form $F = P (d P \land d P) P, (d P \land d P) (X, Y) = d P (X) d P (Y) - d P (Y) d P (X) .$ For any local $C^{2}$ orthonormal frame $V : U \to C^{d \times r}$ with $V^{*} V = I_{r}$ and $P = V V^{*}$ , the connection matrix $A = V^{*} d V$ is a $C^{1}$ anti-Hermitian matrix-valued one-form, and its curvature matrix is $F_{f r} = d A + A \land A = V^{*} F V, (A \land A) (X, Y) = A (X) A (Y) - A (Y) A (X) .$ If $V^{'} = V g$ for a $C^{2}$ map $g : U \to U (r) : = {h \in C^{r \times r} : h^{*} h = h h^{*} = I_{r}}$ , then $A^{'} = g^{- 1} A g + g^{- 1} d g$ and $F_{f r}^{'} = g^{- 1} F_{f r} g$ . Along every $C^{2}$ path, Kato transport is unitary and intertwines the endpoint projectors.

Statement references

C2 projector bundles and Kato transport

Projected connection and Kato transport are well defined

Finite-dimensional vector-valued forms and their exterior derivative

The wedge product of differential forms

Facts & Assumptions

Given: A finite-dimensional smooth manifold $M$ , a $C^{2}$ constant-rank orthogonal projection family $P$ , a local orthonormal frame $V$ for its range, and a $C^{2}$ path in $M$ .

[F1]

C2 projector bundles and Kato transport: $E_{P, x} = ran P (x)$ , $\nabla s = P d s$ , and $K = [P^{'}, P]$ along a path.

[F2]

Projected connection and Kato transport are well defined: the Kato equation has a unique unitary solution and $P (t) W (t) = W (t) P (0)$ .

[F3]

Finite-dimensional vector-valued forms and their exterior derivative: for finite-dimensional vector-valued forms, $d$ is applied componentwise. In this proof, for matrix-valued one-forms $α, β$ , define $(α \land β) (X, Y) = α (X) β (Y) - α (Y) β (X)$ using matrix multiplication.

[F4]

The wedge product of differential forms: the scalar wedge product is the pointwise alternating product of covectors; the displayed matrix-valued formula uses that alternating convention with ordered matrix products.

Proof

1.1givenF1F3F4algebra

Differentiate $P^{2} = P$ in a tangent direction $X$ to obtain $(d P) (X) P + P (d P) (X) = (d P) (X)$ . Multiplying on both sides by $P$ gives $P (d P) (X) P = 0$ . For a $C^{2}$ section $s$ with $P s = s$ , expand $F_{\nabla} (X, Y) s$ by the curvature definition in the Statement. The second-derivative terms cancel by $X (d s (Y)) - Y (d s (X)) - d s ([X, Y]) = 0$ , leaving $P (d P (X) d s (Y) - d P (Y) d s (X))$ . Substituting $d s (X) = d P (X) s + P d s (X)$ and using $P (d P) P = 0$ leaves $P (d P (X) d P (Y) - d P (Y) d P (X)) P s$ . This is $P (d P \land d P) P s$ , proving the projector formula.

2.1F1F3F4step 1.1algebra

In a local frame write a section as $s = V a$ . Then $\nabla (V a) = P d (V a) = V (V^{*} d V a + d a) = V (d a + A a)$ , where $A = V^{*} d V$ . Differentiating $V^{*} V = I_{r}$ gives $(d V)^{*} V + V^{*} d V = 0$ , so $A^{*} = - A$ . Applying $d + A$ twice gives $d A + A \land A$ ; expressing the already computed curvature in this frame gives $F_{f r} = V^{*} F V = d A + A \land A$ .

3.1F3F4step 2.1algebra

For $V^{'} = V g$ , direct differentiation gives $A^{'} = V^{' *} d V^{'} = g^{- 1} A g + g^{- 1} d g$ , since $g^{*} = g^{- 1}$ . The frame-coordinate covariant derivative changes from $d + A$ to $g^{- 1} (d + A) g$ on coefficient sections. Squaring this identity gives $d A^{'} + A^{'} \land A^{'} = g^{- 1} (d A + A \land A) g$ , which is the claimed curvature covariance.

4.1F1F2given∎

For a path the definitions give $W^{'} = [P^{'}, P] W$ ; [F2] proves that its unique solution is unitary and satisfies $P (t) W (t) = W (t) P (0)$ . This is projector parallel transport and uses no Schrödinger evolution.

Remarks

Here $Γ^{2} (E_{P} ∣_{U})$ denotes the $C^{2}$ sections, and $Ω_{C^{1}}^{1} (U; E_{P})$ the $C^{1}$ one-forms with values in the fibers; the lower spaces use the same regularity. The map $Ψ_{V}$ applies $V^{*}$ to the fiber coordinate. The local-frame equation $Ψ_{V} (\nabla s) = (d + A) Ψ_{V} (s)$ is summarized by this commutative diagram. Step 2.1 proves the identity; the diagram records the coordinate change and is not used as a proof.

Projector curvature and frame covariance

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources