Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 4 / b

Table of contents
- Solution b

Solution

The Hodge star operator is defined by

α \land *_{g} β = ⟨ α, β ⟩_{g} vol_{g}

for forms of the same degree. In four dimensions, under

g = e^{2 ψ} g

, the inner product on

p

-forms scales by

e^{- 2 p ψ}

and the volume form scales by

e^{4 ψ}

. Hence

*_{g}_{Ω^{p}} = e^{(4 - 2 p) ψ} *_{g} .

For

p = 2

the exponent vanishes, so the Hodge star on two-forms is conformally invariant.

Solved by gpt-5.6-sol high.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version