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Let-S-R-2-R-3-Sphere-Q1-Find-metric-tensor-g-Q2-Find-Riemann-metric-tensor-R-jkl-i-Q-3-Find-Ricci-tensor-R-Q-4-Find-Ricci-Scalar-R-Christoffel-symbol-first-kind-1-2-

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Question Number 225479 by Lara2440 last updated on 28/Oct/25
Let S;R^2 →R^3 Sphere Q1. Find metric tensor g_(μν) Q2. Find Riemann metric tensor R_(jkl) ^i Q.3 Find Ricci tensor R_(αβ) Q.4 Find Ricci Scalar R Christoffel symbol first kind Γ_(μνσ) =(1/2)(∂_ν ^ g_(μσ) +∂_σ g_(μν) −∂_μ g_(νσ) ) Christoffel symbol second kind Γ_(jk) ^i =g^(il) Γ_(ljk) =(1/2)g^(iσ) (∂_j ^ g_(lk) +∂_k g_(lj) −∂_l g_(jk) ) Riemann metric tensor R_(jkl) ^i =∂_k Γ_(jl) ^i −∂_l Γ_(jk) ^i +Γ_(km) ^i Γ_(jl) ^m −Γ_(lm) ^i Γ_(jk) ^m
$$\mathrm{Let}\:\mathcal{S};\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}^{\mathrm{3}} \:\mathrm{Sphere}\: \\ $$$${Q}\mathrm{1}.\:\mathrm{Find}\:\mathrm{metric}\:\mathrm{tensor}\:\mathrm{g}_{\mu\nu} \: \\ $$$${Q}\mathrm{2}.\:\mathrm{Find}\:\mathrm{Riemann}\:\mathrm{metric}\:\mathrm{tensor}\:{R}_{{jkl}} ^{{i}} \\ $$$${Q}.\mathrm{3}\:\:\mathrm{Find}\:\mathrm{Ricci}\:\mathrm{tensor}\:\mathrm{R}_{\alpha\beta} \\ $$$${Q}.\mathrm{4}\:\:\mathrm{Find}\:\mathrm{Ricci}\:\mathrm{Scalar}\:\mathcal{R} \\ $$$$\mathrm{Christoffel}\:\mathrm{symbol}\:\mathrm{first}\:\mathrm{kind}\: \\ $$$$\Gamma_{\mu\nu\sigma} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\nu} ^{\:} \mathrm{g}_{\mu\sigma} +\partial_{\sigma} \mathrm{g}_{\mu\nu} −\partial_{\mu} \mathrm{g}_{\nu\sigma} \right) \\ $$$$\mathrm{Christoffel}\:\mathrm{symbol}\:\mathrm{second}\:\mathrm{kind} \\ $$$$\Gamma_{{jk}} ^{{i}} =\mathrm{g}^{{il}} \Gamma_{{ljk}} =\frac{\mathrm{1}}{\mathrm{2}}\mathrm{g}^{{i}\sigma} \left(\partial_{{j}} ^{\:} \mathrm{g}_{{lk}} +\partial_{{k}} \mathrm{g}_{{lj}} −\partial_{{l}} \mathrm{g}_{{jk}} \right) \\ $$$$\mathrm{Riemann}\:\mathrm{metric}\:\mathrm{tensor} \\ $$$${R}_{{jkl}} ^{{i}} =\partial_{{k}} \Gamma_{{jl}} ^{{i}} −\partial_{{l}} \Gamma_{{jk}} ^{{i}} +\Gamma_{{km}} ^{{i}} \Gamma_{{jl}} ^{{m}} −\Gamma_{{lm}} ^{{i}} \Gamma_{{jk}} ^{{m}} \\ $$
Answered by Lara2440 last updated on 29/Oct/25
A1. S^� (θ,ρ)=rsin(θ)cos(ρ)e_1 ^� +rsin(θ)sin(ρ)e_2 ^� +rcos(θ)e_3 ^� (∂S^� /∂θ)=rcos(θ)cos(ρ)e_1 ^� +rcos(θ)sin(ρ)e_2 ^� −rsin(θ)e_3 ^� (∂S^� /∂ρ)=−rsin(ρ)sin(θ)e_1 ^� +rsin(θ)cos(ρ)e_2 ^� +0e_3 ^� r_u ^� ∗r_u ^� =(rcos(θ)cos(ρ))^2 +(rcos(θ)sin(ρ))^2 +(rsin(θ))^2 =r^2 r_u ^� ∗r_v ^� =r_v ^� ∗r_u ^� = −r^2 sin(θ)cos(θ)sin(ρ)cos(ρ)+r^2 sin(θ)cos(θ)sin(ρ)cos(ρ) =0 r_v ^� ∗r_v ^� =r^2 sin^2 (θ)sin^2 (ρ)+r^2 sin^2 (θ)cos^2 (ρ) =r^2 sin^2 (θ) g_(μν) = ((( r^2 ),( 0)),(( 0),(r^2 sin^2 (θ))) ) A2. g_(θθ) =r^2 , g_(θρ) or g_(ρθ) =0 , g_(ρρ) =r^2 sin^2 (θ) ∂_θ g_(θθ) =0 , ∂_ρ g_(θθ) =0 ∂_θ g_(θρ) =0 , ∂_ρ g_(θρ) =0 ∂_θ g_(ρθ) =0 , ∂_ρ g_(ρθ) =0 ∂_θ g_(ρρ) =2r^2 sin(θ)cos(θ) , ∂_ρ g_(ρρ) =0 ∴ Γ_(θρρ) =(1/2)(∂_ρ g_(θρ) +∂_ρ g_(θρ) −∂_θ g_(ρρ) )=−r^2 sin(θ)cos(θ) Γ_(ρθρ) =(1/2)(∂_θ g_(ρρ) +∂_ρ g_(ρθ) −∂_ρ g_(θρ) )=r^2 sin(θ)cos(θ) Γ_(ρρθ) =Γ_(ρθρ) =r^2 sin(θ)cos(θ) Γ_(jk) ^i =g^(il) Γ_(ljk) ∴Γ_(ρρ) ^ρ =g^(θθ) Γ_(θρρ) =(1/r^2 )(−r^2 sin(θ)cos(θ))=−sin(θ)cos(θ) Γ_(θρ) ^ρ =g^(ρρ) Γ_(ρθρ) =(1/(r^2 sin(θ)))∙(r^2 sin(θ)cos(θ))=cot(θ) Γ_(θρ) ^ρ =Γ_(ρθ) ^ρ =cot(θ) R_(ijkl) =∂_k Γ_(ijl) −∂_l Γ_(ijk) +Γ_(ikm) Γ_(jl) ^m −Γ_(ilm) Γ_(jk) ^m R_(θρθρ) =r^2 sin^2 (θ) , R_(ρθρ) ^θ =sin^2 (θ) A3. R_(αβ) ^(Ricci) =∂_γ Γ_(αβ) ^γ −∂_β Γ_(αγ) ^γ +Γ_(γλ) ^γ Γ_(αβ) ^λ −Γ_(βλ) ^γ Γ_(αγ) ^λ ∴R_(αβ) ^(Ricci) { ((R_(θθ) =1)),((R_(ρρ) =sin^2 (θ))) :} R_(μν) ^(Ricci) =(1/r^2 )g_(μν)
$${A}\mathrm{1}.\: \\ $$$$\hat {\boldsymbol{\mathcal{S}}}\left(\theta,\rho\right)={r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +{r}\mathrm{cos}\left(\theta\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\frac{\partial\hat {\boldsymbol{\mathcal{S}}}}{\partial\theta}={r}\mathrm{cos}\left(\theta\right)\mathrm{cos}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{r}\mathrm{cos}\left(\theta\right)\mathrm{sin}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −{r}\mathrm{sin}\left(\theta\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\frac{\partial\hat {\boldsymbol{\mathcal{S}}}}{\partial\rho}=−{r}\mathrm{sin}\left(\rho\right)\mathrm{sin}\left(\theta\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +\mathrm{0}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\hat {\boldsymbol{\mathrm{r}}}_{{u}} \ast\hat {\boldsymbol{\mathrm{r}}}_{{u}} =\left({r}\mathrm{cos}\left(\theta\right)\mathrm{cos}\left(\rho\right)\right)^{\mathrm{2}} +\left({r}\mathrm{cos}\left(\theta\right)\mathrm{sin}\left(\rho\right)\right)^{\mathrm{2}} +\left({r}\mathrm{sin}\left(\theta\right)\right)^{\mathrm{2}} \\ $$$$={r}^{\mathrm{2}} \\ $$$$\hat {\boldsymbol{\mathrm{r}}}_{{u}} \ast\hat {\boldsymbol{\mathrm{r}}}_{{v}} =\hat {\boldsymbol{\mathrm{r}}}_{{v}} \ast\hat {\boldsymbol{\mathrm{r}}}_{{u}} = \\ $$$$−{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\mathrm{sin}\left(\rho\right)\mathrm{cos}\left(\rho\right)+{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\mathrm{sin}\left(\rho\right)\mathrm{cos}\left(\rho\right) \\ $$$$=\mathrm{0} \\ $$$$\hat {\boldsymbol{\mathrm{r}}}_{{v}} \ast\hat {\boldsymbol{\mathrm{r}}}_{{v}} ={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)\mathrm{sin}^{\mathrm{2}} \left(\rho\right)+{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)\mathrm{cos}^{\mathrm{2}} \left(\rho\right) \\ $$$$={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right) \\ $$$$\boldsymbol{\mathrm{g}}_{\mu\nu} =\begin{pmatrix}{\:{r}^{\mathrm{2}} }&{\:\:\:\:\:\:\:\mathrm{0}}\\{\:\mathrm{0}}&{{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\end{pmatrix} \\ $$$${A}\mathrm{2}. \\ $$$${g}_{\theta\theta} ={r}^{\mathrm{2}} \:,\:{g}_{\theta\rho} \:\mathrm{or}\:{g}_{\rho\theta} =\mathrm{0}\:,\:{g}_{\rho\rho} ={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right) \\ $$$$\partial_{\theta} {g}_{\theta\theta} =\mathrm{0}\:,\:\partial_{\rho} {g}_{\theta\theta} =\mathrm{0} \\ $$$$\partial_{\theta} {g}_{\theta\rho} =\mathrm{0}\:,\:\partial_{\rho} {g}_{\theta\rho} =\mathrm{0}\: \\ $$$$\partial_{\theta} {g}_{\rho\theta} =\mathrm{0}\:,\:\partial_{\rho} {g}_{\rho\theta} =\mathrm{0} \\ $$$$\partial_{\theta} {g}_{\rho\rho} =\mathrm{2}{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\:,\:\partial_{\rho} {g}_{\rho\rho} =\mathrm{0} \\ $$$$\therefore\:\Gamma_{\theta\rho\rho} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\rho} {g}_{\theta\rho} +\partial_{\rho} {g}_{\theta\rho} −\partial_{\theta} {g}_{\rho\rho} \right)=−{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\:\:\:\:\Gamma_{\rho\theta\rho} =\frac{\mathrm{1}}{\mathrm{2}}\left(\partial_{\theta} {g}_{\rho\rho} +\partial_{\rho} {g}_{\rho\theta} −\partial_{\rho} {g}_{\theta\rho} \right)={r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\:\:\:\:\Gamma_{\rho\rho\theta} =\Gamma_{\rho\theta\rho} ={r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\Gamma_{{jk}} ^{{i}} ={g}^{{il}} \Gamma_{{ljk}} \\ $$$$\therefore\Gamma_{\rho\rho} ^{\rho} ={g}^{\theta\theta} \Gamma_{\theta\rho\rho} =\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\left(−{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\right)=−\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right) \\ $$$$\:\:\:\:\Gamma_{\theta\rho} ^{\rho} ={g}^{\rho\rho} \Gamma_{\rho\theta\rho} =\frac{\mathrm{1}}{{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)}\centerdot\left({r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\right)=\mathrm{cot}\left(\theta\right) \\ $$$$\:\:\:\:\Gamma_{\theta\rho} ^{\rho} =\Gamma_{\rho\theta} ^{\rho} \:=\mathrm{cot}\left(\theta\right) \\ $$$${R}_{{ijkl}} =\partial_{{k}} \Gamma_{{ijl}} −\partial_{{l}} \Gamma_{{ijk}} +\Gamma_{{ikm}} \Gamma_{{jl}} ^{{m}} −\Gamma_{{ilm}} \Gamma_{{jk}} ^{{m}} \\ $$$${R}_{\theta\rho\theta\rho} ={r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)\:,\:\mathrm{R}_{\rho\theta\rho} ^{\theta} =\mathrm{sin}^{\mathrm{2}} \left(\theta\right) \\ $$$${A}\mathrm{3}. \\ $$$${R}_{\alpha\beta} ^{\mathrm{Ricci}} =\partial_{\gamma} \Gamma_{\alpha\beta} ^{\gamma} −\partial_{\beta} \Gamma_{\alpha\gamma} ^{\gamma} +\Gamma_{\gamma\lambda} ^{\gamma} \Gamma_{\alpha\beta} ^{\lambda} −\Gamma_{\beta\lambda} ^{\gamma} \Gamma_{\alpha\gamma} ^{\lambda} \\ $$$$\therefore\mathrm{R}_{\alpha\beta} ^{\mathrm{Ricci}} \begin{cases}{\mathrm{R}_{\theta\theta} =\mathrm{1}}\\{\mathrm{R}_{\rho\rho} =\mathrm{sin}^{\mathrm{2}} \left(\theta\right)}\end{cases} \\ $$$$\mathrm{R}_{\mu\nu} ^{\mathrm{Ricci}} =\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\mathrm{g}_{\mu\nu} \\ $$

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