Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$.


Let $Z\_1,Z\_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential equation:


$$\langle XZ\_i,Y \rangle + \langle X,YZ\_i \rangle =0$$


for all vector fields $X,Y$ on $M$. Here $XZ$ means a derivative with respect to the vector field $X$ of a $\mathbb{R}^3$-valued function $Z$. We will call such $Z\_1,Z\_2$ *infinitesimal bendings*, since the derivative of a one-parameter family of isometric immersions must satisfy this equation.


**Is it true, that the pointwise cross product $M \ni m\mapsto Z\_1(m)\times Z\_2(m)$ is also an infinitesimal bending?**


I don't have any particular reason to believe that it might be true, i couldn't prove it by algebraic manipulations, and not trained to produce non-flat surfaces admitting independent infinitesimal deformations to find a counterexample.



Cross product of two infinitesimal bendings
Let $M$ be a rotationally symmetric cylinder, $Z\_1$ be a vector field tangent to M generating the isometric translation along the cylinder and $Z\_2$ be a vector field also tangent to M generating the isometric rotation around the cylinder. Then $Z\_1\times Z\_2$ is a vector field which radially expanding or shrinking the cylinder, which is probably a counter example.


