Question

I am reading Nakahara's Geometry, topology and physics and I am a bit confused about the non-coordinate basis (section 7.8). Given a coordinate basis at a point on the manifold ∂/∂x^μ we can pick a linear combination of this in order to obtain a new basis e^α=e^μ_α ∂/∂x^μ such that the new basis is orthonormal with respect to the metric defined on the manifold i.e.g(e^α,e^β)=e^μ_αe^ν_βg_μν=δ_αβ (or =η_αβ). I am a bit confused about the way we perform this linear transformation of the basis. Is it done globally? This would mean that at each point we turned the metric into the a diagonal metric, which would imply that the manifold is flat, which shouldn't be possible as the change in coordinates shouldn't affect the geometry of the system (i.e. the curvature should stay the same). So does this mean that we perform the transformation such that the metric becomes the flat metric just at one point, while at the others will also change, but without becoming flat (and thus preserving the geometry of the manifold)? Then Nakahara introduces local frame rotations, which are rotations of these new basis at each point, which further confuses me to why would you do that, once you already obtained the flat metric at a given point. So what is the point of these new transformations as long as we perform the first kind in the first place? Sorry for the long post I am just confused.

Answer 1

In Nakahara's Geometry, Topology and Physics, the non-coordinate basis is obtained by performing a linear transformation of the coordinate basis at each point on the manifold. This transformation does not make the metric diagonal at every point, but rather makes it flat at each point to preserve the geometry of the manifold. Local frame rotations are introduced to account for the general curvature of the manifold.

Step-by-step explanation:

In Nakahara's Geometry, Topology and Physics, the author discusses the concept of a non-coordinate basis in section 7.8. The goal is to find a new basis that is orthonormal with respect to the metric defined on the manifold. This new basis, denoted as e^α=e^μ_α ∂/∂x^μ, is obtained by performing a linear transformation of the coordinate basis at each point on the manifold.

The linear transformation is not done globally, meaning that the metric does not become a diagonal metric at every point. Instead, at each point, the transformation is performed in such a way that the metric becomes the flat metric at that point, while preserving the geometry of the manifold. Therefore, the curvature of the manifold is not affected by the change in coordinates.

In addition to the transformation to obtain an orthonormal basis, Nakahara introduces local frame rotations, which are rotations of the new basis at each point. These local rotations are necessary to account for the general curvature of the manifold and preserve its geometry. So, the point of these local transformations is to account for the curvature of the manifold, while the initial transformation ensures that the metric becomes flat at each point.