- #1

EnigmaticField

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Given a manifold V with connection [itex]\nabla^\rm{V}[/itex], if [itex]\bf{X}[/itex] and [itex]\bf{Y}[/itex] are vector fields on the tangent bundle of V, the torsion at a point [itex]\rm{p}\in \rm{V}[/itex] is defined as [itex]T=\nabla^\rm{V}_\bf{X}\bf{Y}-\nabla^\rm{V}_\bf{Y}\bf{X}-[\bf{X},\bf{Y}][/itex]. In this formula, on the one hand, [itex]\nabla^\rm{V}_\bf{X}\bf{Y}[/itex] is the projection of [itex]\bf{X}\bf{Y}[/itex] into the tangent space of V at p, [itex]\rm{V_p}[/itex], and [itex]\nabla^\rm{V}_\bf{Y}\bf{X}[/itex] is the projection of [itex]\bf{Y}\bf{X}[/itex] into [itex]\rm{V_p}[/itex]; on the other hand, [itex][\bf{X},\bf{Y}]=\bf{X}\bf{Y}-\bf{Y}\bf{X}[/itex] must be tangent to V so should be the tangent component of [itex]\bf{X}\bf{Y}[/itex] minus the tangent component of [itex]\bf{Y}\bf{X}[/itex]. Thus doesn't [itex]\nabla^\rm{V}_\bf{X}\bf{Y}-\nabla^\rm{V}_\bf{Y}\bf{X}=[\bf{X},\bf{Y}][/itex] always hold? Then doesn't the torsion [itex]T[/itex] always vanish?

When V is a hypersurface of another manifold M with connection [itex]\nabla^\rm{M}[/itex] (in this case [itex]\nabla^\rm{V}[/itex] is treated as an induced connection), the above argument can also be understood by the Gauss equations: [itex]\textbf{X}\bf{Y}=\nabla^\rm{M}_\textbf{X}\textbf{Y}=\nabla^\rm{V}_\textbf{X}\textbf{Y}+\rm{B}(\textbf{X},\textbf{Y})\textbf{N}...(1)[/itex]

[itex]\textbf{Y}\textbf{X}=\nabla^\rm{M}_\textbf{Y}\textbf{X}=\nabla^\rm{V}_\textbf{Y}\textbf{X}+\rm{B}(\textbf{Y},\textbf{X})\textbf{N}...(2)[/itex], in which [itex]\rm{B}[/itex] is the second fundamental form with the property [itex]\rm{B}(\bf{X},\bf{Y})=\rm{B}(\bf{Y},\bf{X})[/itex] and [itex]\textbf{N}[/itex] is the unit normal vector field on V. Then from (1) and (2) we can get [itex]\bf{XY}-\bf{YX}=\nabla^\rm{M}_\bf{X}\bf{Y}-\nabla^\rm{M}_\bf{Y}\bf{X}=\nabla^\rm{V}_\bf{X}\bf{Y}-\nabla^\rm{V}_\bf{Y}\bf{X}[/itex], which says that the torsion always vanishes.

So when on Earth can a nonvanishing torsion ever occur?

Or does my argument in the second paragraph only apply to the case when V is embedded in a higher dimensional manifold because only in that case does saying projecting [itex]\bf{X}\bf{Y}[/itex] into [itex]\rm{V_p}[/itex] to get [itex]\nabla^\rm{V}_\bf{X}\bf{Y}[/itex] make sense? If that's the case does that mean a nonvanishing torsion can only occur when V is not embedded in another mainfold, that is, a nonvanihing torsion can only occur to a non-induced connection?