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I'll use $\Omega \in \Omega^2(P,\mathfrak{g})$ to denote the curvature tensor of $\omega$. One way of identifying these two expressions is through Cartan's structure equation $$\Omega = d\omega + \fra …

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space $$T_f C^\infty(M,N) = \ …

Let $f : N \to M$ denote the immersion. Since $f_*TN$ is invariant under $I$ where $I$ is the underlying almost complex structure of $M$, it induces an almost complex structure $I'$ on $N$. Applying N …

Let $f : M \to E$ be a line bundle over an elliptic curve such that $\deg(M)<0$ and let $G = \mathbf{C}^*$ act on $M$ (on the left) by scalar multiplications. The maximal compact subgroup $K$ of $G$ …