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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2022 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2022 • Question 4

Let \(G_0\) and \(G_1\) be Lie groups. Then the product \(G_0\times G_1\) is a Lie group and we let \(p_i:G_0\times G_1\to G_i\) be the projection to each coordinate. Let \(X\) be a vector field on \(G_0\times G_1\) and \(Y_0\) and \(Y_1\) vector fields on \(G_0\) and \(G_1\) such that \(X\) and the \(Y_i\) are \(p_i\)-related. Show that \(X\) if left-invariant if and only if both \(Y_{0}\) and \(Y_{1}\) are left-invariant.

Raw LaTeX

Let \(G_0\) and \(G_1\) be Lie groups. Then the product \(G_0\times G_1\) is a Lie group
and we let \(p_i:G_0\times G_1\to G_i\) be the projection to each coordinate. Let
\(X\) be a vector field on \(G_0\times G_1\) and \(Y_0\) and \(Y_1\) vector fields on \(G_0\) and
\(G_1\) such that \(X\) and the \(Y_i\) are \(p_i\)-related. Show that \(X\) if left-invariant
if and only if both \(Y_{0}\) and \(Y_{1}\) are left-invariant.

Topic Tags

Left-Invariant Vector Fields Lie Groups and Matrix Lie Groups Pullbacks and Wedge Products Vector Fields and Flows