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1. Second Fundamental Form

   Linked via "Weingarten Map"

   The Shape Operator (Weingarten Map)
   The Second Fundamental Form is intrinsically linked to the Shape Operator, $Sp$, also known as the Weingarten Map\, which operates on the tangent space $Tp\Sigma$ at a point $p \in \Sigma$. The Shape Operator is defined such that for any tangent vector $V \in Tp\Sigma$:
   $$Sp(V) = -\nablaV \mathbf{n}$$
   where $\nablaV \mathbf{n}$ is the covariant derivative\ of the unit normal vector field $\mathbf{n}$ along the direction $V$. The shape operator is a self…