Hodge Theory for R- Manifolds
نویسنده
چکیده
Let X be an R-fold, and let π : E −→ X be a real vector bundle, of rank r, equipped with a positive definite symmetric bilinear form. If e1, . . . , er ∈ π −1(X) are orthonormal, then e1 ∧ · · · ∧ er is a non-trivial vector in ∧r E. Proposition: If f1, . . . , fr is any other orthonormal basis for π −1(X), then e1 ∧ · · · ∧ er = ±f1 ∧ · · · ∧ fr. Proof. Note that fi = g · ei for g ∈ O(r), so det(g) = ±1. So, given this data, I get two points in each fiber of ∧r E. Assumption: This is a trivial two fold cover. Choose one connected component and call that section of ∧r E τ . Given this data, I have:
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