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This is comment rather than answer: Please check it, whether it makes sense... Corollary 2.6 page 11 of Free Loop space and homology by J.L Loday says that For any simply connected space, there is …

$M$ be any Riemannian manifold, and $S^1$ is a circle. We can give Manifold structure to $C^\infty(S^1, M)$ modeled on nuclear frechet space. Take $Imm(S^1, M):\{f\in C^\infty(S^1,M): f \text{ is an …

With the help of the Schwarz-Christoffel map, for a given polygon (given angle), we can find some points on the boundary of the upper half plane, such that a particular Schwarz-Christoffel map takes t …

Let $(U,u)$ is a chart for $M$, and $(V,v)$ be a chart for $N$. $u: U\to u(U)\mathbb R^n$ is diffeomorphism. $u(U)$ and $v(V)$ are open subset of $\mathbb R^n$ and $\mathbb R^m$. Then we can identify …