Math 6510 • Differentiable Manifolds

Consider the following vector fields in \(\mathbb {R}^3\): \[ X=\frac {\partial }{\partial x},\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z}. \] Here \(x,y,z\) are the coordinates on \(\mathbb {R}^3\) and \(\frac {\partial }{\partial x}\), \(\frac {\partial }{\partial y}\), \(\frac {\partial }{\partial z}\), are the three standard coordinate vector fields. Show that there is no (nonempty) surface \(M\subset \mathbb {R}^3\) which is tangent to both vector fields at each of its points.