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user12390
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You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space at $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be construtedconstructed with the exponential map of $N$.

You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space at $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be construted with the exponential map of $N$.

You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space at $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be constructed with the exponential map of $N$.

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user12390
user12390
  • 386
  • 2
  • 5

You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space ofat $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be construted with the exponential map of $N$.

You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space of $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be construted with the exponential map of $N$.

You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space at $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be construted with the exponential map of $N$.

Source Link
user12390
user12390
  • 386
  • 2
  • 5

You can find the smooth case in The Convenient Setting of Global Analysis (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.

If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifold. The tangent space of $f\in C^k(M,N)$ is given by $T_fC^k(M,N)=\Gamma_{C^k}(f^*TN)$ where on $\Gamma_{C^k}(f^*TN)$ one has the usual $C^k$-topology. The charts of $C^k(M,N)$ can be construted with the exponential map of $N$.