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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2019 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2019 • Question 5

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 are given by \(U=(y,-x,w,-z)\) and \(V=(w,z,-y,-x)\). Determine whether it is possible to find a smooth embedded two dimensional submanifold \(N\subset S^3\) such that the vector fields \(U\) and \(V\) are tangent to \(N\) at every point of \(N\), i.e., \(U(p),V(p)\in T_pN\) for every \(p\in N\)?

Raw LaTeX

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

Topic Tags

Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Tangent Bundles and Vector Bundles Vector Fields and Flows