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$\newcommand{\M}{M}$ $\newcommand{\N}{N}$ $\newcommand{\TM}{TM}$ $\newcommand{\TN}{TN}$ $\newcommand{\TstarM}{T^*M}$ $\newcommand{\Ga}{\Gamma}$ Let $\M,\N$ be smooth manifolds, $\phi:\M \to \N$ be a …

$\newcommand{\id}{\operatorname{Id}}$ Well, there is a natural way to view this "pullback-symmetry": Exterior derivative commutes with pullbacks: Let $f:M \to N$ be a smooth map, $E$ a vector bundl …

Consider the following functional: $$ E_k(f)=\frac{1}{2}\int_{M} \| \bigwedge^k df\|^2 \text{Vol}_{M}.$$ Theorem: The Euler-Lagrange equation of $E_2$, is $A(\phi)=0$, where $A(\phi) \in \Ga …

$\newcommand{\al}{\alpha}$ $\newcommand{\euc}{\mathcal{e}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ Let $M,N$ be smooth $n$-dimensional Riemannian manifolds (p …

Let $E$ be a smooth vector bundle over a manifold $M$, equipped with a connection $\nabla$. The set of $\nabla$-compatible metrics on $E$ forms a convex cone. This cone can be empty, however (see he …

$\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\tr}{\ope …

This is a cross-post. Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\nabla$, …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\sAverage}[1]{\langle#1\rangle} $ $\new …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\deriv}[2]{\frac{d#1}{d#2}}$ $\newcommand{\sAverage}[1]{\langle#1\rangle} $ $\newcommand{\IP}[2]{\sAverage{#1,#2}}$ $\newcomm …

$\newcommand{\sig}{\sigma}$ $\newcommand{\tr}{\operatorname{tr}_{\eta}}$ $\newcommand{\al}{\alpha}$ $\newcommand{\be}{\beta}$ $\newcommand{\til}{\tilde}$ Let $E$ be a smooth vector bundle over a mani …