Ngô Quốc Anh

October 11, 2009

An example of sectional curvature of sphere

Filed under: Nghiên Cứu Khoa Học, Riemannian geometry — Ngô Quốc Anh @ 15:34

In \mathbb S^n we consider the metric

{g_{ij}} = {\left( {\displaystyle\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}{\delta _{ij}}.

We now find the sectional curvature of g.

Since

{g_{ij}} = \displaystyle{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}{\delta _{ij}}

then

{g^{ij}} = \displaystyle{\left( {\frac{{1 + {{\left| y \right|}^2}}}{2}} \right)^2}{\delta _{ij}}.

Now we need to calculate

\Gamma _{ij}^k = \displaystyle\frac{1}{2}{g^{kl}}\left( {\frac{{\partial {g_{il}}}}{{\partial {y_j}}} + \frac{{\partial {g_{lj}}}}{{\partial {y_i}}} - \frac{{\partial {g_{ij}}}}{{\partial {y_l}}}} \right).

Clearly,

\displaystyle\frac{{\partial {g_{il}}}}{{\partial {y_j}}} = \frac{\partial }{{\partial {y_j}}}\left( {{{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)}^2}{\delta _{il}}} \right) = 2\frac{2}{{1 + {{\left| y \right|}^2}}}\frac{{ - 2}}{{{{\left( {1 + {{\left| y \right|}^2}} \right)}^2}}}\left( {2{y_j}} \right){\delta _{il}} = - 2{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^3}{y_j}{\delta _{il}}.

Similarly, one has the following

\displaystyle\frac{{\partial {g_{lj}}}}{{\partial {y_i}}} = - 2{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^3}{y_i}{\delta _{lj}}, \quad \frac{{\partial {g_{ij}}}}{{\partial {y_l}}} = - 2{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^3}{y_l}{\delta _{ij}}.

Therefore,

\displaystyle \Gamma _{ij}^k= \frac{{ - 2}}{{1 + {{\left| y \right|}^2}}}\left( {{y_j}{\delta _{ik}} + {y_i}{\delta _{kj}} - {y_k}{\delta _{ij}}} \right).

Next we need to calculate coefficients R^m_{lij} of the curvature tensor. To this purpose, we use

\displaystyle R_{lij}^m = \frac{{\partial \Gamma _{lj}^m}}{{\partial {y_i}}} - \frac{{\partial \Gamma _{li}^m}}{{\partial {y_j}}} + \Gamma _{in}^m\Gamma _{jl}^n - \Gamma _{jn}^m\Gamma _{il}^n.

We have

\displaystyle\frac{{\partial \Gamma _{lj}^m}}{{\partial {y_i}}} = \frac{\partial }{{\partial {y_i}}}\left( {\frac{{ - 2}}{{1 + {{\left| y \right|}^2}}}\left( {{y_j}{\delta _{ml}} + {y_l}{\delta _{mj}} - {y_m}{\delta _{lj}}} \right)} \right)

which yields

\displaystyle\frac{{\partial \Gamma _{lj}^m}}{{\partial {y_i}}} ={\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}\left( {{y_i}{y_j}{\delta _{ml}} + {y_i}{y_l}{\delta _{mj}} - {y_i}{y_m}{\delta _{lj}}} \right) + \frac{{ - 2}}{{1 + {{\left| y \right|}^2}}}\left( {{\delta _{ij}}{\delta _{ml}} + {\delta _{il}}{\delta _{mj}} - {\delta _{im}}{\delta _{lj}}} \right).

Similarly, one has

\displaystyle\frac{{\partial \Gamma _{li}^m}}{{\partial {y_j}}} = {\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}\left( {{y_i}{y_j}{\delta _{ml}} + {y_j}{y_l}{\delta _{mi}} - {y_j}{y_m}{\delta _{li}}} \right) + \frac{{ - 2}}{{1 + {{\left| y \right|}^2}}}\left( {{\delta _{ji}}{\delta _{ml}} + {\delta _{jl}}{\delta _{mi}} - {\delta _{jm}}{\delta _{li}}} \right).

Therefore,

\displaystyle\frac{{\partial \Gamma _{lj}^m}}{{\partial {y_i}}} - \frac{{\partial \Gamma _{li}^m}}{{\partial {y_j}}} = - \frac{4}{{1 + {{\left| y \right|}^2}}}\left( {{\delta _{il}}{\delta _{mj}} - {\delta _{im}}{\delta _{lj}}} \right).

On the other hands,

\displaystyle\Gamma _{in}^m\Gamma _{jl}^n - \Gamma _{jn}^m\Gamma _{il}^n = {\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}\left\{ \begin{gathered} {y_i}{y_j}{\delta _{ml}} + {y_i}{y_l}{\delta _{mj}} - {y_i}{y_m}{\delta _{jl}} + \hfill \\ {y_l}{y_j}{\delta _{im}} + {y_j}{y_l}{\delta _{im}} - y_n^2{\delta _{jl}}{\delta _{im}} - \hfill \\ {y_m}{y_j}{\delta _{il}} - {y_m}{y_l}{\delta _{ij}} + {y_m}{y_i}{\delta _{jl}} \hfill \\ \hfill \\ - {y_i}{y_j}{\delta _{ml}} - {y_j}{y_l}{\delta _{mi}} + {y_j}{y_m}{\delta _{il}} - \hfill \\ {y_l}{y_i}{\delta _{jm}} - {y_i}{y_l}{\delta _{jm}} + y_n^2{\delta _{il}}{\delta _{jm}} + \hfill \\ {y_m}{y_i}{\delta _{jl}} + {y_m}{y_l}{\delta _{ij}} - {y_m}{y_j}{\delta _{il}} \end{gathered} \right\}

which yields

\displaystyle\Gamma _{in}^m\Gamma _{jl}^n - \Gamma _{jn}^m\Gamma _{il}^n = {\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}{\left| y \right|^2}\left( {{\delta _{il}}{\delta _{jm}} - {\delta _{jl}}{\delta _{im}}} \right).

Thus,

\displaystyle R_{lij}^m = - \frac{4}{{1 + {{\left| y \right|}^2}}}\left( {{\delta _{il}}{\delta _{mj}} - {\delta _{im}}{\delta _{lj}}} \right) + {\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}{\left| y \right|^2}\left( {{\delta _{il}}{\delta _{jm}} - {\delta _{jl}}{\delta _{im}}} \right) = {\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)^2}\left( {{\delta _{im}}{\delta _{lj}} - {\delta _{il}}{\delta _{mj}}} \right).

Finally, one obtains

\displaystyle K\left( {{e_i},{e_j}} \right) = \frac{{\left\langle {R\left( {{e_i},{e_j}} \right){e_j},{e_i}} \right\rangle }}{{\left\langle {{e_i},{e_i}} \right\rangle \left\langle {{e_j},{e_j}} \right\rangle - {{\left\langle {{e_i},{e_j}} \right\rangle }^2}}} = \frac{{R_{jij}^m\left\langle {{e_m},{e_i}} \right\rangle }}{{{{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)}^4}}} = \frac{{R_{jij}^m{{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)}^2}{\delta _{mi}}}}{{{{\left( {\frac{2}{{1 + {{\left| y \right|}^2}}}} \right)}^4}}} = 1.

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