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From the discussion before the example, we learn that g is a global section of T ∗ M ⊗s T ∗ M. Therefore it makes sense to express g in a coordinate patch as g = gab dxa ⊗ dxb ; so gab is symmetric under a ↔ b. A “principal bundle” is entirely analogous to a vector bundle, where instead of “vector space” we have “Lie group,” and transition functions are now translations in the group. ” This is important in gauge theories. However, since particles are associated to vector bundles deﬁned by representations as just discussed, we will focus on vector bundles exclusively.

Here lies their beauty. If we write θ = f (x, y)dx ∧ dy = θab dxa ∧ dxb (take x1 = x, x2 = y), so that a ∂xb ∂xa ∂xb 7 θxy = −θyx = f /2, then θrθ = θab ∂x ∂r ∂θ = (f /2) ab ∂r ∂θ = (rf /2), and we see that the Jacobian emerges from the anti-symmetry property of diﬀerential forms. More generally, if θ is an n-form on an n-manifold, then θ = f dx1 ∧ · · · ∧ dxn in local coordinates, and in a new coordinate system x, 7Here 12 = − 21 = 1, all others vanishing. n = −1, etc. 5. DIFFERENTIAL FORMS 19 1 n θ = f det ∂x ∂ x dx ∧ · · · ∧ dx , and the Jacobian is automatic.

We can write this metric as the symmetric part of (4/(|z|2 + 1)2 )dz ⊗ dz, where dz = dx + idy, etc. We will have more to say about the anti-symmetric part in future chapters. 4. METRICS, CONNECTIONS, CURVATURE 13 A metric is an inner product on the tangent bundle. If v a ∂a and wb ∂b are two vectors v and w, their inner product is g(v, w) = g(v a ∂a , wb ∂b ) = v a wb gab . ” Then the cotangent vector or one-form, wa dxa , has a natural pairing with v equal to the inner product of v and w. In short, the metric provides an isomorphism between the tangent and the cotangent bundles, exactly as an inner product deﬁnes an isomorphism between V and V ∗ .