atlas[Riemann] - calculation of Riemannian tensor atlas[Ricci] - calculation of Ricci tensor atlas[RicciScalar] - calculation of Ricci scalar Calling Sequence: Riemann(Id) Ricci(Id) RicciScalar(Id) Parameters: Id - variable - corresponding identifier
<Text-field bookmark="info" style="Heading 2" layout="Heading 2">Description:</Text-field> The Riemann procedure allows one to calculate the curvature tensor. The procedure is only available if the curvature 2-forms have been calculated (see atlas[Curvature]). The Ricci procedure allows one to calculate the Ricci tensor. The procedure is only available if the curvature 2-forms (see atlas[Curvature]) have been calculated. The RicciScalar procedure allows one to calculate the Ricci scalar. The procedure is only available if the metric tensor is definite (see atlas[Metric]) and the Ricci tensor has been calculated.
<Text-field bookmark="examples" style="Heading 2" layout="Heading 2">Examples:</Text-field>
<Text-field style="Heading 2" layout="Heading 2">3-dimensional sphere </Text-field> restart: with(atlas): Declare forms: Forms(e[j]=1,xi=1); PCRJI3hpRzYiJkkiZUdGJDYjSSJqR0Yk Declare vectors: Vectors(X,Y,Z,E[j]); PCZJIlhHNiJJIllHRiRJIlpHRiQmSSJFR0YkNiNJImpHRiQ= Declare constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0Yy: Constants(lambda); PCgsJF4jIiIiISIiRiRJI1BpRyUqcHJvdGVjdGVkR0kjX1pHNiJJKENhdGFsYW5HRihJJ2xhbWJkYUdGKg== Declare coframe: Coframe(e[1]=2*d(x)/(1+lambda*(x^2+y^2+z^2)),e[2]=2*d(y)/(1+lambda*(x^2+y^2+z^2)),e[3]=2*d(z)/(1+lambda*(x^2+y^2+z^2))); NyUvJkkiZUc2IjYjIiIiLCQqKCIiI0YoLUkiZEdGJjYjSSJ4R0YmRigsJkYoRigqJkknbGFtYmRhR0YmRigsKCokKUYvRitGKEYoKiQpSSJ5R0YmRitGKEYoKiQpSSJ6R0YmRitGKEYoRihGKCEiIkYoLyZGJTYjRissJCooRitGKC1GLTYjRjhGKEYwRjxGKC8mRiU2IyIiJCwkKihGK0YoLUYtNiNGO0YoRjBGPEYo Declare frame: Frame(E[i]); NyUvJkkiRUc2IjYjIiIiKiYsKiNGKCIiI0YoKihGK0YoSSdsYW1iZGFHRiZGKClJInhHRiZGLEYoRigqKEYrRihGLkYoKUkieUdGJkYsRihGKCooRitGKEYuRigpSSJ6R0YmRixGKEYoRigtSSVEaWZmRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiRJIUdGJkYwRigvJkYlNiNGLComRipGKC1GODYkRj1GM0YoLyZGJTYjIiIkKiZGKkYoLUY4NiRGPUY2Rig= d(x); LCoqJiMiIiIiIiNGJSZJImVHNiI2I0YlRiVGJSoqRiRGJUYnRiVJJ2xhbWJkYUdGKUYlKUkieEdGKUYmRiVGJSoqRiRGJUYnRiVGLEYlKUkieUdGKUYmRiVGJSoqRiRGJUYnRiVGLEYlKUkiekdGKUYmRiVGJQ== Declare metric on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRkFGK0ZGRklGQQ== (see atlas[Metric]): Metric(g=e[1]&.e[1]+e[2]&.e[2]+e[3]&.e[3]); L0kiZ0c2IiwoLUkjJi5HRiQ2JCZJImVHRiQ2IyIiIkYpRiwtRic2JCZGKjYjIiIjRi9GLC1GJzYkJkYqNiMiIiRGNEYs Connection calculation: Connection(omega); Jkkmb21lZ2FHNiI2JEkiaUdGJEkiakdGJA== Curvature calculation: Curvature(Omega); JkkmT21lZ2FHNiI2JEkiaUdGJEkiakdGJA== Riemannian tensor calculation: Riemann(R); L0kiUkc2IiwoKiZJJ2xhbWJkYUdGJCIiIi1JIyYuR0YkNiQtSSMmXkdGJDYkJkkiZUdGJDYjRigmRjA2IyIiI0YsRihGKComRidGKC1GKjYkLUYtNiRGLyZGMDYjIiIkRjhGKEYoKiZGJ0YoLUYqNiQtRi02JEYyRjpGQEYoRig= Ricci tensor calculation: Ricci(r); L0kickc2IiwoKigiIiMiIiJJJ2xhbWJkYUdGJEYoLUkjJi5HRiQ2JCZJImVHRiQ2I0YoRi1GKEYoKihGJ0YoRilGKC1GKzYkJkYuNiNGJ0YzRihGKCooRidGKEYpRigtRis2JCZGLjYjIiIkRjhGKEYo Ricci scalar calculation: RicciScalar(s); L0kic0c2IiwkKiYiIiciIiJJJ2xhbWJkYUdGJEYoRig=
<Text-field style="Heading 2" layout="Heading 2">Example 2</Text-field> restart: with(atlas): Declare forms: Forms(e[j]=1,xi=1); PCRJI3hpRzYiJkkiZUdGJDYjSSJqR0Yk Declare vectors: Vectors(X,Y,Z,E[j]); PCZJIlhHNiJJIllHRiRJIlpHRiQmSSJFR0YkNiNJImpHRiQ= Declare coframe: Coframe(e[1]=x*d(x)+y*d(y),e[2]=x*d(y)-y*d(x)); NyQvJkkiZUc2IjYjIiIiLCYqJkkieEdGJkYoLUkiZEdGJjYjRitGKEYoKiZJInlHRiZGKC1GLTYjRjBGKEYoLyZGJTYjIiIjLCYqJkYrRihGMUYoRigqJkYwRihGLEYoISIi Declare frame: Frame(E[i]); NyQvJkkiRUc2IjYjIiIiLCYqKCwmKiQpSSJ4R0YmIiIjRihGKCokKUkieUdGJkYvRihGKCEiIkYuRigtSSVEaWZmRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiRJIUdGJkYuRihGKCooRitGM0YyRigtRjU2JEY6RjJGKEYoLyZGJTYjRi8sJiooRitGM0YyRihGNEYoRjMqKEYrRjNGLkYoRjxGKEYo Connection definition: omega[1,1]:=x*e[1]; KiYmSSJlRzYiNiMiIiJGJ0kieEdGJUYn omega[2,2]:=y*e[2]; KiZJInlHNiIiIiImSSJlR0YkNiMiIiNGJQ== omega[1,2]:=y*e[1]; KiZJInlHNiIiIiImSSJlR0YkNiNGJUYl omega[2,1]:=-x*e[2]; LCQqJkkieEc2IiIiIiZJImVHRiU2IyIiI0YmISIi Connection declaration: Connection(omega); Jkkmb21lZ2FHNiI2JEkiaUdGJEkiakdGJA== Curvature calculation: Curvature(Omega); JkkmT21lZ2FHNiI2JEkiaUdGJEkiakdGJA== Riemann(R); L0kiUkc2IiwqKiwjIiIiIiIjRihJInlHRiRGKCwoRighIiIqJClJInhHRiQiIiRGKEYoKiZGL0YoKUYqRilGKEYoRigsJiokKUYvRilGKEYoKiRGMkYoRihGLC1JIyYuR0YkNiUmSSJFR0YkNiNGKCZJImVHRiRGPC1JIyZeR0YkNiRGPSZGPjYjRilGKEYsKixGJ0YoRi9GKCwoRjBGLEYtRihGMUYoRihGM0YsLUY4NiUmRjtGQ0Y9Rj9GKEYoKipGJ0YoLChGL0YsKiZGMkYoRjVGKEYoKiQpRioiIiVGKEYoRihGM0YsLUY4NiVGOkZCRj9GKEYoKixGJ0YoRipGKCwoRjBGKEYtRihGMUYoRihGM0YsLUY4NiVGSEZCRj9GKEYo Ricci calculation: Ricci(r); L0kickc2IiwqKipJInlHRiQiIiIsKEYoISIiKiQpSSJ4R0YkIiIkRihGKComRi1GKClGJyIiI0YoRihGKCwmKiQpRi1GMUYoRigqJEYwRihGKEYqLUkjJi5HRiQ2JCZJImVHRiQ2I0YoJkY6NiNGMUYoRioqKkYtRigsKEYuRipGK0YoRi9GKEYoRjJGKi1GNzYkRjlGOUYoRioqKCwoRi1GKiomRjBGKEY0RihGKCokKUYnIiIlRihGKEYoRjJGKi1GNzYkRjxGPEYoRigqKkYnRigsKEYuRihGK0YoRi9GKEYoRjJGKi1GNzYkRjxGOUYoRio= RicciScalar(s); Warning, There is no actual metric tensor
<Text-field bookmark="seealso" style="Heading 2" layout="Heading 2">See Also: </Text-field> atlas, atlas[Connection], atlas[Curvature], atlas[Metric].