Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 115 / 3

If $E$ has local frame transition matrices $g_{ab}$ and the cotangent bundle has transitions $J_{ab}^{-T}$, then $E\otimes {\Lambda }^r{T}^{\ast }B$ has local trivializations with transitionswhich satisfy the cocycle condition. Thus it is a well-defined tensor product of vector bundles.

A connection on a vector bundle is a linear mapsatisfying $\nabla (fs)=df\otimes s+f\nabla s$. Contracting with a vector field gives the covariant derivative ${\nabla }_{X}s$. In a local frame, $\nabla =d+A$ for a matrix-valued one-form $A$; under a frame change $g$ the matrix transforms asIts covariant exterior derivative is defined byand locallyThis formula and the graded Leibniz rule show that definitions in different frames agree.

The curvature form of a connection is $F(A)=d_A^2$. Locally,Its covariant derivative satisfies the Bianchi identityIndeed, substituting $F=dA+A\wedge A$, using $d^2=0$, and applying the graded Leibniz rule leaves equal and opposite terms.

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The induced dual connection is uniquely defined bywhich immediately gives the required pairing identity. Ifthen

A connection on $TB$ is symmetric, or torsion-free, whenequivalently ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$ in coordinates. With $\mathrm{Alt}B(X,Y)=B(X,Y)-B(Y,X)$,

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