Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 4 / e

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- Solution e

Solution

For the stated flat metric and orientation,

* d x^{i} = (- 1)^{i - 1} d x^{1} \land \dots \land d x^{i} \land \dots \land d x^{n} .

Because the metric coefficients and the coordinate one-forms are constant, the Hodge Laplacian acts coefficientwise:

Δ (α_{i} d x^{i}) = (Δ α_{i}) d x^{i} .

Thus a harmonic one-form

α = \sum_{i} α_{i} d x^{i}

has harmonic coefficient functions. Every harmonic function on the compact connected torus is constant by the maximum principle for harmonic functions. Hence

H^{1} (T^{n}) = span_{R} {d x^{1}, \dots, d x^{n}} ≅ R^{n} .

The Hodge decomposition theorem now gives

H_{dR}^{1} (T^{n}) ≅ R^{n} .

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