The Geometry of Physics: An Introduction

2-dimensional 2-sphere axis basis bi-invariant boundary cartesian closed curve coefficients compact complex components compute connection consider constant contravariant coordinate patch coordinate system covariant derivative covector curvature defined deformation diffeomorphism differential Dirac ds² dx² dx³ e₁ electromagnetic equations euclidean example exterior fiber Figure frame function Gauss geodesic given global hermitian homology homomorphism homotopy integral Lagrangian left invariant Lie algebra Lie group line bundle linear transformation Lorentz matrix metric tensor Minkowski space mixed tensor normal Note orientation orthogonal orthonormal p-form parallel displaced parameterized particle plane Problem PROOF Riemannian metric rotation scalar Show simply sphere submanifold subset subspace surface symmetric tangent bundle tangent vector tensor theorem topological torus vanishes variation vector bundle vector field vector space vol³ write yields