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We verify if Gausssian curvature of surfaces and normal curvature of curvesin surfaces introduced by Diniz-Veloso arXiv:1210.7110 and byBalogh-Tyson-Vecchi arXiv:1604.00180 to prove Gauss-Bonnet theorems inHeisenberg space $\mathbb H^1$ are equal. The authors in arXiv:1604.00180utilize a limit of Gaussian and normal curvatures defined in the Riemannianapproximations scheme $(\mathbb R^3,g_L)$ introduced by Gromov to studysub-Riemannian spaces. They show that these limits exist (unlike the limit ofRiemannian surface area form or length form), and they obtain Gauss-Bonnettheorem in $\mathbb H^1$ as limit of Gauss-Bonnet theorems in $(\mathbbR^3,g_L)$ when $L$ goes to infinity. The approach in arXiv:1210.7110 uses ananalogous of Gauss map defined in the unitary cylinder and they show that thecurvature defined by the Gauss map corresponds to the curvature of covariantderivative which is null on left invariant vector fields of $\mathbb H^1$. Theproof of Gauss-Bonnet theorem follows as the classical one. Curvatures do notcoincide. For comparison sake we apply the same formalism of arXiv:1210.7110 toget the curvatures of arXiv:1604.00180 . With the obtained formulas, it ispossible to prove the Gauss-Bonnet theorem in arXiv:1604.00180 as astraightforward application of Stokes theorem.  To exemplify the Gaussian curvature of arXiv:1604.00180 we calculate therotation surfaces of constant curvature in $\mathbb H^1$, which are only ofthree types.

Rotation surfaces of constant Gaussian curvature as Riemannian approximation scheme in sub-Riemannian Heisenberg space $\mathbb H^1$