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3

The statement actually holds in any dimension! There are references to this appearing in Hopf and Alexandroff's Topologie (1935), but I'm sure it was known earlier. Claim. Every compact, connected, smooth manifold admits a vector field with at most one singularity. Proof. By compactness and genericity, we can find a vector field with finitely many ...

2

Was it "not a surface" or "not a smooth surface" or "not a closed surface"? This is the surface of rotation of the graph of $y = \cos z$ about the $z$-axis. It is singular at z= $\frac \pi 2 + k\pi$. But other than that it qualifies as a surface.

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This question remains confusingly stated. I agree with @studiosus that the intent of the question is to ask this: If the intrinsic Gaussian curvature of $N$ is equal to the extrinsic Gaussian curvature of $N$ (at every point), then must the ambient manifold be flat, i.e., a Riemannian manifold of constant sectional curvature $0$? The answer is that this ...

2

What you want is the area formula from multivariable calculus. In your case each of the three surfaces $S_1,S_2,S_3$ is the graph of a function $z=f(x,y)$ defined for all $(x,y)$ in some (measurable) subset $D$ of the plane. Any (measurable) subset of the surface can be written in the form $f(A)$ for some (measurable) subset $A \subset D$. Furthermore, ...

1

The second one appears to belong to a family of surfaces defined by Banchoff based on the Chmutov surfaces of order $n$. Explicitly, the implicit equation is given by $$3+8(x^4+y^4+z^4) = 8(x^2+y^2+z^2).$$ The first one is not known to me. It appears to have a tetrahedral symmetry.

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Well known Beltrami- Enneper theorem. May be treated in: A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart Normal curvature vanishes, total curvature is entirely in tangent plane, so that geodesic torsion is $$\tau_g = \sqrt {- K}$$

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