Second-order Laplace-type operators on metric vector bundles

Let $M$ be a Hausdorff second-countable smooth manifold without boundary of dimension $n\ge 2$ (Smooth manifolds and their smooth charts), let $\mu$ be a smooth positive density on $M$ (Density bundle and smooth density fields), and let $E\to M$ be a finite-rank smooth vector bundle (Smooth vector bundles, rank, fibres, and trivial bundles) over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ carrying a smooth bundle metric $⟨\cdot ,\cdot ⟩$ (Smooth bundle metrics), Euclidean in the real case and Hermitian, linear in the first variable, in the complex case; in the complex case the fibres are complex vector spaces through a smooth complex structure on $E$ (Complex-linear and metric-compatible bundle connections). Write $\mathrm{\Gamma }(E)$ for the smooth sections (Smooth sections, local sections, and support).

A $\mathbb{K}$-linear map $P:\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E)$ is a linear differential operator of order at most two on $E$ when, on every chart domain $\mathrm{\Omega }$ of $M$ with coordinates $x=(x^1,\dots ,x^n)$ and in every local frame of $E$ over $\mathrm{\Omega }$ in which a section is written $s\leftrightarrow u=(u^1,\dots ,u^r)$ and its image as $Ps\leftrightarrow ((Ps{)}_1,\dots ,(Ps{)}_r)$, $(Ps{)}_{\lambda }=-{\sum }_{i,j=1}^n{a}_{\lambda \mu }^{ij}(x){\mathrm{\partial }}_i{\mathrm{\partial }}_j{u}^{\mu }(x)+{\sum }_{j=1}^n{b}_{\lambda \mu }^j(x){\mathrm{\partial }}_j{u}^{\mu }(x)+c_{\lambda \mu }(x)u^{\mu }(x),1\le \lambda \le r,$ with smooth coefficient functions on $\mathrm{\Omega }$ and summation over the repeated index $\mu$ (and over $i,j$). Such an operator is Laplace type when in addition, in every such chart and frame, its second-order coefficients are scalar multiples of the identity, $a_{\lambda \mu }^{ij}=a^{ij}{\delta }_{\lambda \mu }$, with $a^{ij}$ smooth real functions for which the matrix $(a^{ij}(x))$ is symmetric and positive definite at every $x\in \mathrm{\Omega }$. Its principal symbol is the endomorphism-valued quadratic function ${\sigma }_2(P)(x,\xi )=a^{ij}(x){\xi }_i{\xi }_j{\mathrm{i}\mathrm{d}}_{E_x}$ for $\xi \in T_x^{\ast }M$. Its scalar principal form $a$ is a smooth section of ${\mathrm{S}\mathrm{y}\mathrm{m}}^{2}TM$ (A smooth tensor field), since it acts on two covectors, and $a_x(\xi ,\xi )>0$ for $\xi \mathrm{\ne }0$. For positive-rank $E$ the symbol determines $a$ uniquely. On the zero-rank bundle the operator and its symbol are zero; a supplied smooth positive-definite form $a$ is part of the Laplace-type data in this degenerate case.

For the density-integral convention assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). $P$ is formally symmetric for $(\mu ,⟨\cdot ,\cdot ⟩)$ when ${\int }_M⟨Ps,t⟩d\mu ={\int }_M⟨s,Pt⟩d\mu$ for all compactly supported smooth sections $s,t\in {\mathrm{\Gamma }}_c(E)$ (Integral of a compactly supported smooth density).

Well-definedness. The definition is local. On a fixed chart and frame the displayed assignment is a $\mathbb{K}$-linear map on components with smooth coefficients, and the chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$) computes the transition to another chart and frame on the overlap: the second-order coefficients transform by the symmetric tensor rule $a^{\mathrm{\prime }kl}=a^{ij}\frac{\mathrm{\partial }{x}^{\mathrm{\prime }k}}{\mathrm{\partial }{x}^i}\frac{\mathrm{\partial }{x}^{\mathrm{\prime }l}}{\mathrm{\partial }{x}^j}$ together with conjugation by the frame-change matrix, while the terms produced by the second derivatives of the coordinate change are absorbed into the first-order coefficients. Hence a single operator that admits a local expression on one cover admits a local expression on every chart and frame, and the two hypotheses placed on the leading coefficient are invariant: scalar-ness is preserved by frame conjugation and by the coordinate rule for a field of symmetric matrices, and symmetric positive definiteness is preserved by both. For positive rank the scalar principal form, and in every rank the principal symbol, are therefore independent of the chart and frame, since two local expressions of the same operator differ by an operator of order at most one; the symbol is smooth in $(x,\xi )$ because the $a^{ij}$ are smooth, and the scalar principal form is positive on nonzero covectors exactly when $(a^{ij})$ is positive definite. Formal symmetry is an identity between integrals of compactly supported smooth sections and therefore depends only on the operator, the density and the metric.

• Choice. The local symbol computations use supplied data and no choice principle. Countable Choice is inherited only by the stated density-integral convention. No existence of densities, metrics or partitions is asserted.
• Sign convention. The minus sign in the leading term makes the model operator $-\mathrm{\Delta }$ on functions Laplace type with principal symbol $\mid \xi {\mid }^2$; the invariance statement above does not depend on this sign choice, which is fixed once and for all for this page and its companion.