Affinor geometry on Lie algebroids (Q2799786)

If \(M\) is a paracompact orientable manifold of class \(C^\infty\), \(E\) is a real vector bundle of rank \(r\) over \(M\), \(\pi\) is a projection \(E\to M\), and \(\alpha\) is a fiber 1-form on \(E\), then the vector bundle \(E\) is called a Lie algebroid if the Lie bracket \([\;,\;]\) is given on the set \(C^\infty(E)\) of all sections and there is a linear mapping \(\Psi:C^\infty(E)\to C^\infty(TM)\) satisfying \([\Psi\sigma,\Psi\tau]=\Psi[\sigma,\tau]\) and \([\sigma,f\tau]=(\Psi\sigma)(f)\tau+f[\sigma,\tau]\) for any \(f\in C^\infty(M)\) and any sections \(\sigma,\tau\in C^\infty(E)\). \(\Psi\) defines a homomorphism between \(E\) and \(TM\) and determines the action of a section \(\sigma\) on \(f\) by \(\sigma(f)=(\Psi\sigma)(f)\). On \(E\), there exists an exterior differential \(d\) defined as \(d\alpha(\sigma,\tau)=\sigma(\alpha(\tau))-\tau(\alpha(\sigma))-\alpha([\sigma,\tau])\). The set \(\text{rad}\,\alpha\) of all sections \(\sigma:i_\sigma d\alpha=0\), where \(i\) denotes the inner product, is called the radical of \(\alpha\). A 1-form \(\alpha\) is called regular if its radical has a constant dimension at all points of \(M\). The radical of a regular non-closed 1-form is a vector subbundle of \(E\) of rank \(\leq r-2\). Since \(E\) is always equipped with a Riemannian metric \(g\), then it is assumed that \(g\) defines a fiber scalar product \((\;,\;)\) on \(E\). Hence, \(E=\text{rad}\,\alpha\oplus D\) is the orthogonal sum of \(\text{rad}\,\alpha\) and the orthogonal vector subbundle \(D\) called the working bundle. If \(\Phi\) is a continuous endomorphism field of \(E\) satisfying \((\Phi\sigma,\Phi\tau)=(\sigma,\tau)\) and \(d\,\alpha(\sigma,\tau)=(\Phi\sigma,\tau)\) for all sections \(\sigma\) and \(\tau\), then \(\Phi\) satisfies:NEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[(i)] \(\Phi^2_{|D}=-\text{id}\), NEWLINE\item[(ii)] \(\Phi^*d\,\alpha=d\,\alpha\), NEWLINE\item[(iii)] \(\text{rad}\,\alpha=\ker\Phi\), NEWLINE\item[(iv)] \(\Phi^*=-\Phi\), NEWLINE\item[(v)]\(\phi\) is positive definite for all \(x\in M\).NEWLINENEWLINE\end{itemize}} NEWLINEFor a Lie algebroid \(E\) with a regular 1-form \(\alpha\) and a working vector bundle \(D\), a continuous field \(\Phi\) of endomorphisms of \(E\) is called an affinor associated with \(\alpha\) if it satisfies (i)--(iii) and (vi)\,\(d\,\alpha(\sigma,\Phi\sigma)\geq 0\) for any section \(\sigma\). A symmetric 2-form \(\beta\) on \(E\) is called a radical metric if \(\text{rad}\,\beta=D\) and \(\beta(\sigma,\sigma)\geq 0\) for all sections \(\sigma\). An affinor structure on a Lie algebroid \(E\) is a pair \((\alpha,\Phi)\) if \(\alpha\) is a regular 1-form on \(E\) and \(\Phi\) is an affinor associated with \(\alpha\).NEWLINENEWLINEIn this paper, the author studies properties of affinor structures. It is shown that the radical metric \(\beta\) on \(E\) is of the form \(\beta=\exp(f)\alpha\otimes\alpha\). Also, the author shows that where \(\alpha\) is a regular 1-form on a real Lie algebroid \(E\) and \(D\) is the working subbundle of \(E\), then there is a bijection between the set of all affinors associated with \(\alpha\) and the set of complex structures on \(D\) associated with the 2-form \(d\,\alpha\). Finally, it is shown that if on a Lie algebroid \(E\) there exists an affinor structure with working bundle \(D\), then \(e(\Lambda^2(E))=W_1(D)=0\), where \(e(E)\) is the Euler class of \(E\), \(\Lambda^2(E)\) is a vector bundle of exterior 2-forms on \(E\), and \(W_1(D)\) is the first Stiefel-Whitney class of \(D\).NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].