Manifolds

Source: https://www.euclideanspace.com/maths/geometry/space/surfaces/manifold/index.htm

• Like a surface in $n$-dimensions (hypersurface)
• An $n$-dim manifold looks like $\mathbb{R}^n$ locally (locally Euclidian)
  • Circle: 1-dim manifold. If we zoom around a point on the circle, it looks like a line ($\mathbb{R}^1$)
  • Sphere: 2-dim manifold. Zooming onto a point, it looks like a plane ($\mathbb{R}^2$)

Source: https://www.seas.upenn.edu/~meam620/slides/kinematicsI.pdf

An $n$-dim manifold is a a set $M$ which is locally homeomorphic to $\mathbb{R}^n$

• Homeomorphism: a map $f$ from $M$ to $N$ and its inverse are both continuous
• Smooth map: all partial derivatives of $f$, of all orders, exist and are continuous
• Diffeomorphism: smooth map and with all partial derivatives of $\text{inv}(f)$, of all orders, exist and are continuous

A group that is a differentiable manifold is called a Lie group .