Generalized Pythagorean Flow Conjecture: For any n-dimensional Riemannian manifold M, there exists a notion of "generalized Pythagorean angle" between tangent vectors, defined intrinsically in terms of the Riemannian metric, such that:
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The generalized Pythagorean angle agrees with the Euclidean angle when M is Euclidean space.
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The generalized Pythagorean angle is related to the lengths of the sides of geodesic triangles on M in a way that generalizes the Pythagorean theorem. Specifically, for any geodesic triangle on M with sides of length a, b, and c, and generalized Pythagorean angles α, β, and γ opposite these sides, we have:
f(a, α) + f(b, β) + f(c, γ) = 0
for some function f that reduces to the Euclidean Pythagorean theorem when M is Euclidean. -
The generalized Pythagorean angle can be defined in one of the following ways: a) As the angle between two tangent vectors with respect to a suitable Hermitian metric on the complexified tangent bundle of M, generalizing the notion of angle from complex geometry. b) In terms of parallel transport of tangent vectors along geodesics on M, in a way that depends only on the intrinsic geometry of M.
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For any Pythagorean (n+1)-tuple (a₁, a₂, ..., aₙ, aₙ₊₁), there exists a geometric flow on M that minimizes a global measure of the deviation of the generalized Pythagorean angles from the "90-degree angles" defined by the tuple. This measure is called the generalized Pythagorean angle defect.
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If the generalized Pythagorean angle defect is initially finite, then the flow exists for all time and converges to a metric g₀ that realizes the "90-degree angles" of the Pythagorean tuple in the sense of the generalized Pythagorean angles.
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The limit metric g₀ is unique up to isometry, and it is a stable fixed point of the flow.
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If M is compact, then the convergence to g₀ occurs exponentially fast in a suitable norm.