Using the Gauss-Bonnet Theorem

Question

Let $M$ be a closed, connected and oriented surface that is immersed in the sphere $\mathbb{S}^3$. Let $K$ denote the Gauss curvature of $M$ (the product of the principal curvatures) and denote by $K_{sec}$ the sectional curvature of $\mathbb{S}^3$ restricted to the tangent planes of $M$. I want to apply Gauss-Bonnet's Theorem to $M$. Should I integrate $K_{sec}$? That is, the following formula is true? $$ \int_M K_{sec} \, \operatorname{d} \operatorname{vol}_M = 2 \pi \chi(M)$$

I suspect this is the case, because Gauss-Bonnet is intrinsic, whereas $K$ is extrinsic.

Answer 1

No, you need the Gaussian curvature of $M$ with the induced metric. (This is only going to be the product of the principal curvatures when $M$ is embedded in a flat space. Consider, for example, a great $2$-spheres in $S^3$, whose second fundamental form is identically $0$.) $K_{\text{sec}}=1$ for all $2$-planes in $T_pS^3$.

EDIT: There are classical formulas for the (intrinsic) curvature of a Riemannian surface in terms of its first fundamental form. This is also very nicely calculated using orthonormal moving frames to get the connection $1$-form $\omega_{12}$ and the intrinsic Gaussian curvature $K$ is given by the equation $-K\,dA = d\omega_{12}$. Indeed, using this equation, it's not hard to give a proof of the Gauss-Bonnet theorem using Stokes's Theorem.

In general, if $M\subset N$ is a submanifold of the Riemannian manifold $N$, then the Gauss equations relate the curvature of $M$ to the second fundamental form and the curvature tensor of $N$. In particular, when $N=S^3$ has constant curvature $1$, the formula you want is $$ K = k_1k_2 + 1.$$ Note that this checks with the basic examples we were discussing in comments.

Answer 2

Ted Shifrin already gave the explanation. You can deduce it from definition. Choose a local O.N. frame $\{e_1,e_2\}$ on $M$, with a normal vector $N$, it gives a O.N. frame for $S^3$. By definition \begin{eqnarray*} 1=K_{sec}=\tilde{K}(e_1,e_2)&=&<\tilde{R}(e_1,e_2)e_1,e_2>\\ &=& <R(e_1,e_2)e_1,e_2>+h_{12}^2-h_{11}h_{22}\\ &=& K_M-k_1k_2 \end{eqnarray*} Where we choose $e_i$ to be the principal directions. So $K_M=k_1k_2+1$ is the Guassian curvature of $M$ with the induced metric. Only in flat space you can say the Guassian curvature is the product of principal curvatures, but in our case, the ambient manifold is constantly curved space.