I have a problem with a computation which shows that the sectional curvature coincide with the Gauss Curvature in dimension 2. This is the definition of sectional curvature I am using:
$K_{XY}(p)=-\frac{R(X,Y,X,Y)}{||X||^2 ||Y||^2 – <X,Y>^2}$
Conceptually I understood that the previous one is a generalization of the Gauss curvature, but I don't understand how to recover the Gauss curvature from the sectional curvature, in particular I do not see how $-R(E_1,E_2,E_1,E_2)=eg-f^2$ where $eg-f^2$ is the determinant of the second fundamental form, and $E_1,E_2$ are the coordinate vector fields.
Best Answer
This is an interesting question. Jack gives a hint to the solution, but let me fill in the details.
What we want to show is that how $R(X,Y,X,Y)$ is related to the second fundamental form.The idea is regard the surface $M$ as a submanifold of $\mathbb R^3$, note that here the metric $g$ of $M$ is induced by the canonical metric of $\mathbb R^3$. Now, the Levi-Civita connection $\nabla$ on $(M,g)$ satisfies $$(\nabla_XY)(p)=(\nabla_X^EY)(p)^{\top}$$
where $(\nabla^E)$ is the Levi-Civita connection on $\mathbb R^3$, and $v^{\top}$ means projection onto the tangent space of $M$. Let's introduce Gauss's equation in the following theorem:
Gauss Theorem: The curvature tensor $R$ of a submanifold $M\subset \mathbb R^n$ is given by the Gauss equation $$\langle R(X,Y)W,Z\rangle=\alpha(X,Z)\cdot \alpha(Y,W)-\alpha(X,W)\cdot \alpha(Y,Z)$$
Where $\alpha(X,Y)=(\nabla_X^EY)(p)-(\nabla_X^EY)(p)^{\top}=(\nabla_X^EY)(p)^{\bot}$, i.e., projecting vector onto normal space, and $(\nabla^E)$ is the canonical Levi-Civita connection on $\mathbb R^n$. One can see [this reference] 1 for details in section 1.8~1.10.
Specifically, in our case, $M$ is a two dimensional surface in $\mathbb R^3,$ where we have the parameterization $r=r(u,v)$, with spanning vector fields of tangent bundle $\{r_u, r_v\}$. Also in $\mathbb R^3$, the normal space of $M$ at a point is spanned by a unit normal vector $n$, and we can write $\alpha(X,Y)=\nabla_X^EY\cdot n$. And by Gauss theorem, we have $$-\langle R(r_u,r_v)r_u,r_v\rangle=\alpha(r_u,r_u)\cdot \alpha(r_v,r_v)-\alpha(r_u,r_v)\cdot \alpha(r_v,r_u)$$
Now the second fundamental form is given by $L=r_{uu}\cdot n, M= r_{uv}\cdot n$, and $N=r_{vv}\cdot n$ (note here I use different notation of second fundamental form as yours). $$\alpha(r_u,r_u)=\nabla_{r_u}^Er_u\cdot n=\frac{\partial}{\partial u}r_u\cdot n=r_{uu}\cdot n=L,$$ and the others expressions are similar, thus we have $$-\langle R(r_u,r_v)r_u,r_v\rangle=LM-N^2$$ which gives us the second fundamental form as we desired.