Evaluate sphere tensor components with clear coordinate inputs. Check metric terms, Christoffel symbols, and curvature invariants fast. Export clean results for geometry study and verification.
| Radius a | θ | Unit | gθθ | gφφ | Rθφθφ | K | Ricci Scalar |
|---|---|---|---|---|---|---|---|
| 3 | 45 | deg | 9.000000 | 4.500000 | 4.500000 | 0.111111 | 0.222222 |
| 2 | 60 | deg | 4.000000 | 3.000000 | 3.000000 | 0.250000 | 0.500000 |
| 5 | 1.2 | rad | 25.000000 | 21.718220 | 21.718220 | 0.040000 | 0.080000 |
For a sphere of radius a, the standard metric in coordinates (θ, φ) is:
ds² = a² dθ² + a² sin²θ dφ²
From this metric:
gθθ = a²
gφφ = a² sin²θ
Γθφφ = -sinθ cosθ
Γφθφ = Γφφθ = cotθ
The main curvature components become:
Rθφθφ = sin²θ
Rθφθφ = a² sin²θ
K = 1 / a²
Ricci scalar = 2 / a²
This means the sphere has constant positive curvature everywhere.
The result section appears below the header and above the form after submission.
The Riemann curvature tensor measures how geometry bends. For a sphere, the tensor shows constant positive curvature. This makes the sphere a classic object in differential geometry. Students often study it before moving to curved surfaces and general relativity.
This calculator finds metric terms, Christoffel symbols, Riemann tensor entries, Ricci tensor terms, Gaussian curvature, and Ricci scalar. These values come from the standard spherical coordinate chart. The page gives both component form and scalar curvature invariants.
A 2-sphere uses coordinates θ and φ. The radius stays fixed. Only the angular directions vary. Because of that, the metric becomes simple but still curved. The first metric term stays constant. The second changes with sin²θ. This reflects shrinking circles near the poles.
The sphere has the same curvature at every point. That is why Gaussian curvature equals 1 divided by radius squared. The Ricci scalar becomes 2 divided by radius squared. Larger spheres curve less. Smaller spheres curve more. This clean pattern helps learners verify manual derivations.
This tool is useful for homework checking, lecture review, and symbolic intuition. You can test different radii and angles quickly. The results also help compare local coordinate expressions with global geometric meaning. Export features make it easier to save examples and share worked outputs.
At the poles, cotθ becomes undefined in these coordinates. That issue comes from the coordinate chart, not the sphere itself. The surface remains smooth. When reading values, focus on both tensor components and curvature invariants. Together they give a clearer picture of intrinsic geometry.
It computes metric components, Christoffel symbols, selected Riemann tensor entries, Ricci tensor terms, Gaussian curvature, and Ricci scalar for a 2-sphere.
Sphere curvature scales as 1 divided by radius squared. A larger sphere is flatter locally, while a smaller sphere bends more strongly.
Some metric and tensor components contain sinθ or cotθ. Their numerical values depend on the selected coordinate point.
Yes. The sphere has constant positive Gaussian curvature. Some coordinate components vary with θ, but the intrinsic curvature stays uniform.
The spherical coordinate chart becomes singular at θ = 0 and θ = π. That is a coordinate issue, not a geometric defect of the sphere.
The calculator shows Rθφθφ = a² sin²θ. This is a standard nonzero covariant Riemann component for the sphere.
Yes. The input allows both angle units. The calculator converts degrees to radians before evaluating the trigonometric expressions.
You can download a CSV file for data work and generate a PDF from the visible result block for printing or sharing.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.