On the canonical extension of a differentiable manifold (Q1841877)

In this paper, the differential geometry of second canonical extension \(^2M\) of a differential manifold \(M\) is studied. If \(M\) is an \(m\)-dimensional manifold, the second canonical extension \[ ^2M=\bigl\{A\mid A\in TTM,\;\pi_{TM}(A)=d\pi_M(A)\bigr\} \] of \(M\), where \(\pi_M: TM\to M\) and \(\pi_{TM}: TTM\to TM\) are the canonical projections, is a \(3m\)-dimensional differential manifold. Some vector fields tangent to \(^2M\) in \(TTM\) are determined. First, if \(X\in \chi(M)\), then the second order complete lift \(X^{cc}\) of \(X\) to \(TTM\) is tangent to \(^2M\). For a differentiable manifold \(M\) with a linear connection \(\widehat \nabla\) let \(\widehat \nabla^{cc}\) be the complete lift of \(\widehat \nabla^c\) defined by \[ \widehat \nabla^{cc}_{\overline X^c} \overline Y^c= (\widehat \nabla^c_{\overline X} \overline Y)^c. \] Thus, it is proved that if \(^2X\), \(^2Y\in \chi(^2M)\) that the vector field \(\widehat \nabla^{cc}_{d \iota(^2X)} d\iota(^2Y)\) is tangent to \(^2M\), where \(\iota:{^2M}\to TTM\) is the natural inclusion. In addition, the authors obtain that the second canonical extensions of \(M\) and a totally geodesic submanifold in \(M\) are totally geodesic submanifolds in \(TTM\) and \(^2M\) respectively.