Classical manifold

Prove that the set:

$$\mathbb{S}=\left \{ x \in \mathbb{R}^n \bigg|\;\; \left \| x \right \|_{2}=1 \right \}$$

defines a manifold.

Solution

Position vector perpendicular to the derivative

Let $ a: I \rightarrow \mathbb{R}^3 $ , where $ I $ is an open interval of the real line endowed with the usual topology, be a smooth , parametrized curve that does not pass through the origin (that is the point $O(0, 0, 0) $ ) . If $ a(t_0) $ is the shortest to the origin point of the curve $ a$ and $ a'(t_0) \neq 0 $ , then prove that the position vector $ a(t_0) $ is perpendicural to $ a'(t_0) $.

Solution

Change of variables

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a differentiable function. If $\{y_1, \dots, y_n \}$ are the coordinates of the image of $f$ and $\{x_1, \dots, x_n\}$ are the coordinates of the domain , prove that:

$$f^{\star}\,\omega=\left(\det\,\mathrm{d}f\right)\,\mathrm{d}x_1\,\land...\land\,\mathrm{d}x_{n}$$

where $\omega= \mathrm{d}y_1\,\land...\land\,\mathrm{d}y_{n}$.

Solution

Constant sign

Let $f:\mathbb{S} \subset \mathbb{R}^3 \rightarrow \mathbb{R}$ be a non zero continuous function on a connected surface $\mathbb{S}$. Prove that $f$ does not change sign on $\mathbb{S}$.

Solution

Isoperimetric inequality

Let $\gamma$ be a simple closed curved of length $\ell$ enclosing an area $A$. Prove that:

$$4\pi A \leq \ell^2$$

Solution:

Klein Bottle

In continuity of Möbius strip we present in this post the Klein Bottle surface.
First of all we give a visualization.

The line is the shortest path between two points

Let $p, q$ be two points of $\mathbb{R}^n$, $u$ be a unit vector and let $\gamma$ be a curve passing through those points. ($\gamma(a)=p, \; \gamma(b)=q$). Prove that the shortest path between these two points is the line.

Solution:

Möbius strip

Prove that the Möbius strip is not oriented but its boundary is closed and positive oriented.

Solution:

We kick things off by giving the visualization of  Möbius strip.