Gauss's Theorema Egregium (intrinsic Gauss curvature)

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Problem 3.2.26 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.2 "Differential Geometry of Surfaces"

(a)(b)(c)(d)

Let $M$ be a 2-dimensional manifold smoothly embedded in $\mathbb {R}^{3}$ with unit normal $n$ .

(a)

Explain what is meant by intrinsic and extrinsic quantities on $M$ .

(b)

Are the principal curvatures intrinsic?

(c)

Discuss why the covariant derivative on $M$ , defined using the covariant derivative on $\mathbb {R}^{3}$ , is intrinsic.

(d)

Assuming (c), discuss why the Gauss curvature of $M$ is intrinsic.

Problem 3.2.30 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.2 "Differential Geometry of Surfaces"

Show that the ellipsoid $x^2+2y^2+3z^2=1$ is not isometric to any sphere $x^2+y^2+z^2=r^2$ .

Problem 3.2.38 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.2 "Differential Geometry of Surfaces"

(a)(b)

Let $X \subset \mathbb {R}^{3}$ be the cone $x^2=y^2+z^2$ , and let $Y$ be the torus $(\sqrt{x^2+y^2}-2)^2+z^2=1$ , that is, the torus obtained by rotating the circle $(x-2)^2+z^2-1=y=0$ around the $z$ -axis.

(a)

Show that for any point $p \in X$ other than the vertex $(0,0,0)$ , there is a neighborhood of $p$ in $X$ isometric to an open subset of the Euclidean plane $\mathbb {R}^{2}$ .

(b)

Show that no open subset of $Y$ is isometric to any open subset of the Euclidean plane.

Problem 3866 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.2 "Finding Functions from Their Total Differential"

The differential of the arc length on some surface is expressed by the formula $ds^{2}=A(x,y)\, dx^{2}+B(x,y)\, dy^{2}$ . Introducing new variables by the formulas $u=P(x,y)$ , $v=Q(x,y)$ , where $P$ and $Q$ are some functions, the formula for $ds^{2}$ can be given the form $ds^{2}=du^{2}+dv^{2}$ if the functions $P$ and $Q$ satisfy the conditions

\left(\frac{\partial P}{\partial x}\right)^{2}+\left(\frac{\partial Q}{\partial x}\right)^{2}=A(x,y), \quad \left(\frac{\partial P}{\partial y}\right)^{2}+\left(\frac{\partial Q}{\partial y}\right)^{2}=B(x,y),

\frac{\partial P}{\partial x}\frac{\partial P}{\partial y}+\frac{\partial Q}{\partial x}\frac{\partial Q}{\partial y}=0.

Prove that, for such $P$ and $Q$ to be selectable, the functions $A(x,y)$ and $B(x,y)$ must satisfy the condition

\frac{\partial \left(\frac{1}{\sqrt{A}}\frac{\partial \sqrt{B}}{\partial x}\right)}{\partial x}+\frac{\partial \left(\frac{1}{\sqrt{B}}\frac{\partial \sqrt{A}}{\partial y}\right)}{\partial y}=0.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.