I load the following file
theManifold = Manifold(1,'M',r'\mathcal{M}')
theManifold = Manifold(1,'M',r'\mathcal{M}')
theOpenSet = theManifold.open_subset('U')
firstChart.<x> = theOpenSet.chart(r'x')
g=theManifold.riemannian_metric('g')
g[0,0] = x
show("g with first chart")
show(g.display())
secondChart.<y> = theOpenSet.chart('y')
first_to_second = firstChart.transition_map(secondChart,[2*x])
second_to_first = first_to_second.inverse()
show("g with secondChart.frame() and secondChart as arguments to display")
show(g.display(secondChart.frame(),secondChart))
theManifold.set_default_frame(secondChart.frame())
theManifold.set_default_chart(secondChart)
show("g with secondChart.frame() and secondChart as defaults")
show(g.display())
The output is
𝚐 𝚠𝚒𝚝𝚑 𝚏𝚒𝚛𝚜𝚝 𝚌𝚑𝚊𝚛𝚝
g=x dx⊗dx
𝚐 𝚠𝚒𝚝𝚑 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝.𝚏𝚛𝚊𝚖𝚎() 𝚊𝚗𝚍 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝 𝚊𝚜 𝚊𝚛𝚐𝚞𝚖𝚎𝚗𝚝𝚜 𝚝𝚘 𝚍𝚒𝚜𝚙𝚕𝚊𝚢
g=1/8 y dy⊗dy
𝚐 𝚠𝚒𝚝𝚑 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝.𝚏𝚛𝚊𝚖𝚎() 𝚊𝚗𝚍 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝 𝚊𝚜 𝚍𝚎𝚏𝚊𝚞𝚕𝚝𝚜
g=1/4 x dy⊗dy
In particular, when the secondChart and secondChart.frame() have been set as defaults, the metric g is expanded with respect to the second frame, but the components are still expressed in the first coordinates.
Is there a way to set defaults so that g appears as 1/8 y dy ⊗dy?
