📂Geometry

Homotopy to Null in Differential Geometry

Definition¹

Let’s say that a closed curve $\gamma$ encloses a region $\mathscr{R}$ on a surface $M$. Suppose $\sigma$ is a closed curve or a loop with period $L$ placed on $\mathscr{R}$. And let $\sigma (0) = x_{0}$. If there exists a closed curve $\sigma_{s}$ on the surface $M$ that satisfies the following conditions for $s \in [0,1]$, then $\sigma$ is said to be null-homotopic.

• $\sigma_{s}(0) = x_{0}$
• $\sigma_{0}(t) = \sigma (t)$ and $\sigma_{1}(t) = x_{0}$
• $\sigma_{s}(t) \in \mathscr{R} \quad \forall s\in [0,1], t\in(0, L)$
• The following function $\Gamma$ is continuous. $$\Gamma : [0,1] \times [0, L] \to M \text{ given by } \Gamma (s,t) = \sigma_{s}(t)$$

Explanation

In other words, saying that $\sigma$ is null-homotopic means that $\sigma$ can continuously transform on $\mathscr{R}$ into a single point $x_{0}$.