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For a Riemannian submersion with totally geodesic submanifolds as fibers, the positive Laplace-Beltrami operator commutes with pullback of functions. Horizontal terms in the Riemannian Hessian are pulled back from the base, and vertical terms vanish. Minimal fibers suffice because only the trace of the vertical second fundamental form enters.

If a unit tangent makes angle $\theta$ with the first principal direction, the normal curvature is ${\kappa }_n={\kappa }_1{\cos }^2\theta +{\kappa }_2{\sin }^2\theta$. This follows by diagonalizing the second fundamental form relative to the first fundamental form. When the Gaussian curvature is nonnegative, $|{\kappa }_n|\le 2|H|$.

The Gauss–Codazzi equations relate the intrinsic curvature and connection of an embedded submanifold to its second fundamental form and the ambient curvature. The Gauss equation controls tangential curvature; the Codazzi equation controls the covariant derivative of the second fundamental form. The Gauss formula is the tangential-normal splitting used to derive these relations.

For a Euclidean embedded submanifold,It expresses intrinsic curvature in terms of the second fundamental form.

For a submanifold with induced Levi-Civita connection $\nabla$ and second fundamental form $A$, the ambient derivative decomposes asFor a parametrized surface this reads $X_{ij}={\Gamma }_{ij}^k{X}_k+h_{ij}N$.

Every regular cone parametrized by $X(r,t)=rc(t)$ has zero Gaussian curvature away from its vertex. Its unit normal is independent of $r$, whileis tangent, so two coefficients of the second fundamental form vanish and $\det II=0$.