Question Number 225488 by Lara2440 last updated on 30/Oct/25
$$\mathrm{Diffenrantial}\:\mathrm{Geometry}…..=\wedge= \\ $$$$\mathrm{Christoffel}\:\mathrm{symbol}\: \\ $$$$\Gamma_{\sigma\mu\nu} \:\mathrm{first}\:\mathrm{kind}\:\mathrm{and}\:\Gamma_{\mu\nu} ^{\:\sigma} \:\mathrm{second}\:\mathrm{kind}… \\ $$$$\mathrm{and}\:\mathrm{Chritoffel}\:\mathrm{symbol}\:\mathrm{satisfy} \\ $$$$\Gamma_{\mu\nu} ^{\:\sigma} =\frac{\mathrm{1}}{\mathrm{2}}{g}^{\sigma{l}} \Gamma_{{l}\mu\nu} \\ $$$$\Gamma_{{ijk}} \:\:\left\{\theta,\rho\right\}\in\left\{{i},{j},{k}\right\} \\ $$$$\Gamma_{\theta\theta\theta} \:,\:\Gamma_{\rho\rho\rho} \:,\:\Gamma_{\theta\theta\rho} \\ $$$$\Gamma_{\theta\rho\theta} \:,\:\Gamma_{\rho\theta\rho} \:,\:\Gamma_{\rho\theta\theta} \\ $$$$\Gamma_{\theta\rho\rho} \:,\:\Gamma_{\rho\rho\theta} \\ $$$$\Gamma_{{jk}} ^{{i}} =\frac{\mathrm{1}}{\mathrm{2}}{g}^{{il}} \Gamma_{{ljk}} \\ $$$${g}_{\mu\nu} =\begin{pmatrix}{\:{r}^{\mathrm{2}} }&{\:\:\:\:\:\:\mathrm{0}}\\{\:\mathrm{0}}&{{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\end{pmatrix} \\ $$$$\left({g}_{\mu\nu} \right)^{−\mathrm{1}} ={g}^{\mu\nu} \\ $$$${g}^{\mu\nu} =\begin{pmatrix}{\:\frac{\mathrm{1}}{{r}^{\mathrm{2}} }}&{\:\:\:\:\:\:\:\:\mathrm{0}}\\{\:\mathrm{0}}&{\frac{\mathrm{1}}{{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)}}\end{pmatrix} \\ $$$${g}_{\theta\theta} ={r}^{\mathrm{2}} ,\:{g}_{\theta\rho} =\mathrm{0}\:,\:{g}_{\rho\theta} =\mathrm{0}\:,\:{g}_{\rho\rho} ={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right) \\ $$$$\partial_{\theta} {g}_{\theta\theta} =\mathrm{0}\:,\:\partial_{\rho} {g}_{\theta\theta} =\mathrm{0}\:,\:\partial_{\theta} {g}_{\theta\rho} =\mathrm{0}\:,\:\partial_{\rho} {g}_{\theta\rho} =\mathrm{0} \\ $$$$\partial_{\theta} {g}_{\rho\theta} =\mathrm{0}\:,\:\partial_{\rho} {g}_{\rho\theta} =\mathrm{0}\:,\:\partial_{\theta} {g}_{\rho\rho} =\mathrm{2}{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\:,\:\partial_{\rho} {g}_{\rho\rho} =\mathrm{0} \\ $$$$\Gamma_{{abc}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{{b}} ^{\:} {g}_{{ac}} +\partial_{{c}} {g}_{{ab}} −\partial_{{a}} {g}_{{bc}} \right) \\ $$$$\Gamma_{\theta\theta\theta} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\theta} {g}_{\theta\theta} +\partial_{\theta} {g}_{\theta\theta} −\partial_{\theta} {g}_{\theta\theta} \right)=\mathrm{0} \\ $$$$\Gamma_{\rho\rho\rho} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\rho} {g}_{\rho\rho} +\partial_{\rho} {g}_{\rho\rho} −\partial_{\rho} {g}_{\rho\rho} \right)=\mathrm{0} \\ $$$$\Gamma_{\theta\rho\rho} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\rho} {g}_{\theta\rho} +\partial_{\rho} {g}_{\theta\rho} −\partial_{\theta} {g}_{\rho\rho} \right)=−{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\Gamma_{\rho\theta\rho} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\theta} {g}_{\rho\rho} +\partial_{\rho} {g}_{\rho\theta} −\partial_{\rho} {g}_{\theta\rho} \right)={r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\Gamma_{\rho\theta\theta} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\theta} {g}_{\rho\theta} +\partial_{\theta} {g}_{\rho\theta} −\partial_{\rho} {g}_{\theta\theta} \right)=\mathrm{0} \\ $$$$\Gamma_{\theta\rho\theta} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\rho} {g}_{\theta\theta} +\partial_{\theta} {g}_{\theta\rho} −\partial_{\theta} {g}_{\rho\theta} \right)=\mathrm{0} \\ $$$$\Gamma_{\theta\theta\rho} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\theta} {g}_{\theta\rho} +\partial_{\rho} {g}_{\theta\theta} −\partial_{\theta} {g}_{\theta\rho} \right)=\mathrm{0} \\ $$$$\Gamma_{\rho\rho\theta} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\rho} {g}_{\rho\theta} +\partial_{\theta} {g}_{\rho\rho} −\partial_{\rho} {g}_{\rho\theta} \right)={r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\Gamma_{{jk}} ^{{i}} ={g}^{{il}} \Gamma_{{ljk}} \\ $$$${g}^{\theta\theta} =\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\:,\:{g}^{\theta\rho} =\mathrm{0}\:,\:{g}^{\rho\theta} =\mathrm{0}\:,\:{g}^{\rho\rho} =\frac{\mathrm{1}}{{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)} \\ $$$$\Gamma_{\theta\theta} ^{\theta} ={g}^{\theta{l}} \Gamma_{{l}\theta\theta} =\begin{cases}{{g}^{\theta\theta} \Gamma_{\theta\theta\theta} =\mathrm{0}}\\{{g}^{\theta\rho} \Gamma_{\rho\theta\theta} =\mathrm{0}}\end{cases},\:\Gamma_{\rho\theta} ^{\theta} ={g}^{\theta{l}} \Gamma_{{l}\rho\theta} =\begin{cases}{{g}^{\theta\theta} \Gamma_{\theta\rho\theta} =\mathrm{0}}\\{{g}^{\theta\rho} \Gamma_{\rho\rho\theta} =\mathrm{0}}\end{cases}\:,\:\Gamma_{\theta\rho} ^{\theta} ={g}^{\theta{l}} \Gamma_{{l}\theta\rho} =\begin{cases}{{g}^{\theta\theta} \Gamma_{\theta\theta\rho} =\mathrm{0}}\\{{g}^{\theta\rho} \Gamma_{\rho\theta\rho} =\mathrm{0}}\end{cases}\:,\:\Gamma_{\rho\rho} ^{\theta} ={g}^{\theta{l}} \Gamma_{{l}\rho\rho} =\begin{cases}{{g}^{\theta\theta} \Gamma_{\theta\rho\rho} =−\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)}\\{{g}^{\theta\rho} \Gamma_{\rho\rho\rho} =\mathrm{0}}\end{cases} \\ $$$$\Gamma_{\theta\theta} ^{\rho} ={g}^{\rho{l}} \Gamma_{{l}\theta\theta} =\begin{cases}{{g}^{\rho\theta} \Gamma_{\theta\theta\theta} =\mathrm{0}}\\{{g}^{\rho\rho} \Gamma_{\rho\theta\theta} =\mathrm{0}}\end{cases}\:,\:\Gamma_{\rho\theta} ^{\rho} ={g}^{\rho{l}} \Gamma_{{l}\rho\theta} \:=\begin{cases}{{g}^{\rho\theta} \Gamma_{\theta\rho\theta} =\mathrm{0}}\\{{g}^{\rho\rho} \Gamma_{\rho\rho\theta} =\mathrm{cot}\left(\theta\right)}\end{cases},\:\Gamma_{\theta\rho} ^{\rho} ={g}^{\rho{l}} \Gamma_{{l}\theta\rho} =\begin{cases}{{g}^{\rho\theta} \Gamma_{\theta\theta\rho} =\mathrm{0}}\\{{g}^{\rho\rho} \Gamma_{\rho\theta\rho} =\mathrm{cot}\left(\theta\right)}\end{cases}\:,\:\Gamma_{\rho\rho} ^{\rho} ={g}^{\rho{l}} \Gamma_{{l}\rho\rho} =\begin{cases}{{g}^{\rho\theta} \Gamma_{\theta\rho\rho} =\mathrm{0}}\\{{g}^{\rho\rho} \Gamma_{\rho\rho\rho} =\mathrm{0}}\end{cases} \\ $$$$\therefore\Gamma_{\rho\rho} ^{\theta} =−\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\:\:\:\Gamma_{\mu\nu} ^{\:\rho} =\Gamma_{\nu\mu} ^{\rho} \\ $$$$\:\:\:\Gamma_{\rho\theta} ^{\rho} =\Gamma_{\theta\rho} ^{\rho} =\mathrm{cot}\left(\theta\right) \\ $$$$\:{R}_{{jkl}} ^{{i}} =\partial_{{k}} \Gamma_{{jl}} ^{{k}} −\partial_{{l}} \Gamma_{{jk}} ^{{i}} +\Gamma_{{km}} ^{{i}} \Gamma_{{jl}} ^{{m}} −\Gamma_{{lm}} ^{{i}} \Gamma_{{jk}} ^{{m}} \\ $$$${R}_{\theta\theta\theta} ^{\theta} \:,\:{R}_{\theta\rho\theta} ^{\theta} \:,\:{R}_{\rho\theta\theta} ^{\theta} \:,\:{R}_{\rho\rho\theta} ^{\theta} \:,\:{R}_{\theta\rho\rho} ^{\theta} \:,\:{R}_{\rho\rho\rho} ^{\theta} \:,\:{R}_{\rho\theta\rho} ^{\theta} \:,\:{R}_{\theta\rho\rho} ^{\theta} \\ $$$${R}_{\theta\theta\theta} ^{\rho} \:,\:{R}_{\theta\rho\theta} ^{\rho} \:,\:{R}_{\rho\theta\theta} ^{\rho} \:,\:{R}_{\rho\rho\theta} ^{\rho} \:,\:{R}_{\theta\rho\rho} ^{\rho} \:,\:{R}_{\rho\rho\rho} ^{\rho} \:,\:{R}_{\rho\theta\rho} ^{\rho} \:,\:{R}_{\theta\rho\rho} ^{\rho} \: \\ $$$$\mathrm{and}\:\mathrm{Riemann}\:\mathrm{metric}\:\mathrm{tensor}\:\mathrm{have}\:\mathrm{symmetries} \\ $$$${R}_{{abcd}} =−{R}_{{bacd}} \\ $$$${R}_{{abcd}} =−{R}_{{abdc}} \\ $$$${R}_{{abcd}} ={R}_{{cdab}} \\ $$$$\mathrm{Non}-\mathrm{Zero}\:\Gamma_{\mu\nu} ^{\:\rho} \: \\ $$$$\Gamma_{\rho\rho} ^{\:\theta} =−\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\:,\:\Gamma_{\rho\theta} ^{\:\rho} \:,\:\Gamma_{\theta\rho} ^{\:\rho} =\mathrm{cot}\left(\theta\right) \\ $$$$\left\{{i},{j},{k},\ell\right\}\in\left\{\theta,\rho\right\} \\ $$$${R}_{\rho\theta\rho} ^{\theta} =\partial_{\theta} \Gamma_{\rho\rho} ^{\theta} −\partial_{\rho} \Gamma_{\rho\theta} ^{\theta} +\Gamma_{\theta{m}} ^{\:\theta} \Gamma_{\rho\rho} ^{{m}} −\Gamma_{\rho{m}} ^{\theta} \Gamma_{\rho\theta} ^{{m}} \\ $$$$=\mathrm{sin}^{\mathrm{2}} \left(\theta\right) \\ $$$${R}_{\alpha\rho\theta\rho} =\mathrm{g}_{\alpha\mu} {R}_{\rho\theta\rho} ^{\mu} \\ $$$${R}_{\theta\rho\theta\rho} =\mathrm{g}_{\theta\mu} {R}_{\rho\theta\rho} ^{\mu} =\begin{cases}{{g}_{\theta\theta} {R}_{\rho\theta\rho} ^{\theta} ={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\\{{g}_{\theta\rho} {R}_{\rho\theta\rho} ^{\rho} =\mathrm{0}}\end{cases} \\ $$$${R}_{\theta\rho\theta\rho} ={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right) \\ $$$$\therefore\:{R}_{\rho\theta\rho} ^{\theta} =\mathrm{sin}^{\mathrm{2}} \left(\theta\right)\:,\:{R}_{\theta\rho\theta\rho} ={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right) \\ $$$$\sim\mathrm{2}-\mathrm{dimensional}\:\mathrm{Riemann}\:\mathrm{Manifold}\sim \\ $$$${R}_{{abcd}} ={K}\left({g}_{{ac}} {g}_{{bd}} −{g}_{{ad}} {g}_{{bc}} \right) \\ $$$${R}_{\mu\nu} ^{\mathrm{Ricci}} ={Kg}_{\mu\nu} \\ $$$${K}\:\mathrm{is}\:\mathrm{gaussian}\:\mathrm{curvature}\:\:\mathrm{aka}\:{K}=\frac{\mathrm{det}\mathbb{II}_{{p}} }{\mathrm{det}\:\mathbb{I}_{{p}} }=\frac{{LM}−{N}^{\mathrm{2}} }{{EG}−{F}^{\mathrm{2}} } \\ $$$${Q}\mathrm{225479} \\ $$