i am trying to find difference among curvatures such as ricci, sectional, scalar,
what is the difference between ricci curvature and ricci curvature tensor? i confused. Is there any way to visualized them?
[Math] What are the ricci curvature
differential-geometry
Best Answer
I think it might help to understand what Gaussian curvature is. The Gaussian curvature of a surface in $\mathbb{R}^3$ gives concrete geometrical meaning to how 'curved' a surface is.
For a geometric interpretation of the sectional curvature, a very good description is given in probably most introduction to curvature or Riemannian geometry books. You can prove that the sectional curvature of a subspace of $T_pM$ is the Gaussian curvature of a corresponding 'totally geodesic' 2-dimensional Riemannian submanifold of $M$. This submanifold is formed by flowing along the geodesics with initial velocities in the above subspace.
Let $K(X,Y)$ denote the sectional curvature of the plane spanned by $X$ and $Y$. Given $V\in T_pM$, if you orthonormally extend $V$ to a basis $\{V,E_2,\dots,E_n\}$ to $T_pM$, you can show that $Ric(V,V)=\sum_{k=2}^nK(V,E_k)$. That is, the Ricci curvature is the sum of Gaussian curvatures of planes spanned by $V$ and elements of an orthonormal basis.
You can also show $S=\sum_{i\not=j}K(E_i,E_j)$. So the scalar curvature is the sum of Gaussian curvatures of planes formed by pairs of elements in the orthonormal basis.
The geometric meanings of Gaussian curvature give a geometric meaning to sectional, Ricci and scalar curvature.
All of this I learned from Lee's Riemannian Manifolds; Intro to Curvature. Giving that a look might help.
I should also add that Ricci curvature = Gaussian Curvature =$\frac{1}{2}$scalar curvature on a $2$ dimensional Riemannian manifold.