I'm trying to understand the manifold properties of world-sheets in string theory. I'm told that world sheets are manifolds and that manifolds are locally Euclidean. So I would like to know the characteristics between the space-time coordinates of the world-sheet given as x(adsbygoogle = window.adsbygoogle || []).push({}); ^{μ}verses the 2D surface parameterized by (σ,τ). Are x^{μ}locally Euclidean? Are the coordinates (σ,τ) locally Euclidean? Remember x^{μ}are functions of the parameters (σ,τ) or x^{μ}=x^{μ}(σ,τ) which defines a surface in space-time. How does this all relate to manifold theory?

Thanks.

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# World-sheets, manifolds, and coordinate systems

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