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Project: test
Views: 117
Kernel: SageMath 7.5
H2 = Manifold(2, 'H^2', start_index=1)
H2
2-dimensional differentiable manifold H^2
X.<x,y> = H2.chart("x y:(0,+oo)") X
Chart (H^2, (x, y))
%display latex
X
(H2,(x,y))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(H^2,(x, y)\right)
g = H2.metric('g')
H2.default_chart()
(H2,(x,y))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(H^2,(x, y)\right)
H2.default_frame()
(H2,(x,y))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(H^2, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y }\right)\right)
g[1,1] = 1/y g[2,2] = 1/y g.display()
g=1ydxdx+1ydydy\renewcommand{\Bold}[1]{\mathbf{#1}}g = \frac{1}{y} \mathrm{d} x\otimes \mathrm{d} x + \frac{1}{y} \mathrm{d} y\otimes \mathrm{d} y
Riem = g.riemann() Riem
Riem(g)\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Riem}\left(g\right)
print(Riem)
Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold H^2
Riem.display()
Riem(g)=12y2xdydxdy+12y2xdydydx+12y2ydxdxdy12y2ydxdydx\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Riem}\left(g\right) = -\frac{1}{2 \, y^{2}} \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} x\otimes \mathrm{d} y + \frac{1}{2 \, y^{2}} \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} y\otimes \mathrm{d} x + \frac{1}{2 \, y^{2}} \frac{\partial}{\partial y }\otimes \mathrm{d} x\otimes \mathrm{d} x\otimes \mathrm{d} y -\frac{1}{2 \, y^{2}} \frac{\partial}{\partial y }\otimes \mathrm{d} x\otimes \mathrm{d} y\otimes \mathrm{d} x
Riem.display_comp()
Riem(g)xyxyxyxy=12y2Riem(g)xyyxxyyx=12y2Riem(g)yxxyyxxy=12y2Riem(g)yxyxyxyx=12y2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \mathrm{Riem}\left(g\right)_{ \phantom{\, x} \, y \, x \, y }^{ \, x \phantom{\, y} \phantom{\, x} \phantom{\, y} } & = & -\frac{1}{2 \, y^{2}} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, x} \, y \, y \, x }^{ \, x \phantom{\, y} \phantom{\, y} \phantom{\, x} } & = & \frac{1}{2 \, y^{2}} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, y} \, x \, x \, y }^{ \, y \phantom{\, x} \phantom{\, x} \phantom{\, y} } & = & \frac{1}{2 \, y^{2}} \\ \mathrm{Riem}\left(g\right)_{ \phantom{\, y} \, x \, y \, x }^{ \, y \phantom{\, x} \phantom{\, y} \phantom{\, x} } & = & -\frac{1}{2 \, y^{2}} \end{array}
Riem[1,2,1,2]
12y2\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2 \, y^{2}}
Riem[:]
[[[[0,0],[0,0]],[[0,12y2],[12y2,0]]],[[[0,12y2],[12y2,0]],[[0,0],[0,0]]]]\renewcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\left[\left[0, 0\right], \left[0, 0\right]\right], \left[\left[0, -\frac{1}{2 \, y^{2}}\right], \left[\frac{1}{2 \, y^{2}}, 0\right]\right]\right], \left[\left[\left[0, \frac{1}{2 \, y^{2}}\right], \left[-\frac{1}{2 \, y^{2}}, 0\right]\right], \left[\left[0, 0\right], \left[0, 0\right]\right]\right]\right]
nabla = g.connection() nabla
g\renewcommand{\Bold}[1]{\mathbf{#1}}\nabla_{g}
nabla(g) == 0
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}
g.christoffel_symbols_display()
Γxxyxxy=12yΓyxxyxx=12yΓyyyyyy=12y\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \Gamma_{ \phantom{\, x} \, x \, y }^{ \, x \phantom{\, x} \phantom{\, y} } & = & -\frac{1}{2 \, y} \\ \Gamma_{ \phantom{\, y} \, x \, x }^{ \, y \phantom{\, x} \phantom{\, x} } & = & \frac{1}{2 \, y} \\ \Gamma_{ \phantom{\, y} \, y \, y }^{ \, y \phantom{\, y} \phantom{\, y} } & = & -\frac{1}{2 \, y} \end{array}