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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1998 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 1998 • Question 3

Suppose the three sphere \[ S^3=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2+z^2+w^2=1\} \] is embedded into four space the standard way. Two independent vector fields on \(S^3\) are given by \(V=(y,-x,w,-z)\), \(W=(w,z,-y,-x)\). Determine whether it is possible to embed a closed smooth two dimensional surface \(N\subset S^3\) so that the vector fields are tangent to \(N\) at every point of \(N\), i.e. \(V(p),W(p)\in T_pN\) for every \(p\in N\).

Raw LaTeX

Suppose the three sphere
\[
  S^3=\{(x,y,z,w)\in\mathbb{R}^4:x^2+y^2+z^2+w^2=1\}
\]
is embedded into four space the standard way. Two independent vector fields on \(S^3\)
are given by \(V=(y,-x,w,-z)\), \(W=(w,z,-y,-x)\). Determine whether it is possible to
embed a closed smooth two dimensional surface \(N\subset S^3\) so that the vector
fields are tangent to \(N\) at every point of \(N\), i.e. \(V(p),W(p)\in T_pN\) for every
\(p\in N\).

Topic Tags

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