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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2005 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2005 • Question 1

In this problem we identify \(\mathbb {C}\) with \(\mathbb {R}^2\) in the usual manner via \[ z\mapsto (\operatorname {Re}z,\operatorname {Im}z). \] Consider the following smooth submanifolds of \(\mathbb {C}^2-\{(0,0),(0,0.5)\}\): \[ M_1=\{(z,w)\in \mathbb {C}^2\mid z^2+w^2=1\} \] \[ M_2=\{(z,w)\in \mathbb {C}^2\mid z^2-w^2+w=1\} \] \[ M_3=\{(x_1,y_1,x_2,y_2)\in \mathbb {R}^4\mid 1-x_1+x_2^2+y_1^2+y_2^2=0\}. \] Which of the pairs \(M_i,M_j\) (\(i,j\in \{1,2,3\}\)) are transverse at \(p=(1,0)\in \mathbb {C}^2\)?

Raw LaTeX

In this problem we identify \(\mathbb{C}\) with \(\mathbb{R}^2\) in the usual manner
via
\[
  z\mapsto(\operatorname{Re}z,\operatorname{Im}z).
\]
Consider the following smooth submanifolds of
\(\mathbb{C}^2-\{(0,0),(0,0.5)\}\):
\[
  M_1=\{(z,w)\in\mathbb{C}^2\mid z^2+w^2=1\}
\]
\[
  M_2=\{(z,w)\in\mathbb{C}^2\mid z^2-w^2+w=1\}
\]
\[
  M_3=\{(x_1,y_1,x_2,y_2)\in\mathbb{R}^4\mid
  1-x_1+x_2^2+y_1^2+y_2^2=0\}.
\]
Which of the pairs \(M_i,M_j\) (\(i,j\in\{1,2,3\}\)) are transverse at
\(p=(1,0)\in\mathbb{C}^2\)?

Topic Tags

Smooth Manifolds and Embedded Submanifolds Transversality and Intersection Theory