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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1999 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 1999 • Question 5

Consider the two vector fields in \(\mathbb {R}^3\) given by \(V(x,y,z)=(x^2+y^2,0,-y)\) and \(W(x,y,z)=(0,x^2+y^2,x)\). Find a two dimensional surface \(M^2\subset \mathbb {R}^3\) containing the point \(p=(1,0,0)\) which is tangent to the vector fields \(V\) and \(W\) in a neighborhood of \(p\), or show that such a surface \(M\) doesn’t exist.

Raw LaTeX

Consider the two vector fields in \(\mathbb{R}^3\) given by
\(V(x,y,z)=(x^2+y^2,0,-y)\) and \(W(x,y,z)=(0,x^2+y^2,x)\). Find a two dimensional
surface \(M^2\subset\mathbb{R}^3\) containing the point \(p=(1,0,0)\) which is tangent to
the vector fields \(V\) and \(W\) in a neighborhood of \(p\), or show that such a surface
\(M\) doesn't exist.

Topic Tags

Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups Vector Fields and Flows