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Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2023 • Question 4

Question

Math 6510 • Differentiable Manifolds Spring 2023 • Question 4

Find two 1-forms \(\alpha \) and \(\beta \) on \(\mathbb {R}^2-\{(0,0),(0,2)\}\) that are closed but not exact and have the property that \(\alpha +t\beta \) is not exact for any \(t\in \mathbb {R}\). (Hint: Let \(\iota :S^1\hookrightarrow \mathbb {R}^2\) be inclusion of \(S^1\) into \(\mathbb {R}^2\). Then choose \(\alpha \) and \(\beta \) such that

\[ \int _{S^1}\iota ^*(\alpha +t\beta )=\int _{S^1}\iota ^*\alpha \ne 0.) \]

Raw LaTeX

Find two 1-forms \(\alpha\) and \(\beta\) on \(\mathbb{R}^2-\{(0,0),(0,2)\}\) that are closed but
not exact and have the property that \(\alpha+t\beta\) is not exact for any \(t\in\mathbb{R}\).
(Hint: Let \(\iota:S^1\hookrightarrow\mathbb{R}^2\) be inclusion of \(S^1\) into \(\mathbb{R}^2\). Then choose \(\alpha\)
and \(\beta\) such that

\[
\int_{S^1}\iota^*(\alpha+t\beta)=\int_{S^1}\iota^*\alpha\ne 0.)
\]

Topic Tags

de Rham Cohomology