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An $n$-dimensional smooth manifold is a second countable Hausdorff space $M^n$ together with a collection of maps called "charts" such that:

• a chart is a homeomorphism $\phi : U\to U'$ where $U$ is open in $M^n$ and $U'$ is open in $\boldsymbol{R}^n$;
• each point $x\in M$ is in the domain of some chart;
• for charts $\phi : U\to U'$ and $\psi: V\to V'$ we have that the "change of coordinates" $\phi\psi^{-1} : \psi(U\cap V) \to \phi(U\cap V)$ is $C^\infty$; and
• the collection of charts is maximal with these properties.
This definition is given in Bredon (1993). If $M$ is a smooth manifold with an inner product $\langle \cdot, \cdot\rangle_x$ on the tangent space $T_xM$ for every $x\in M$, and $\langle \cdot,\cdot\rangle_x$ varies smoothly with respect to $x$, then $M$ is called a Riemannian manifold. If, in addition, $M$ is connected, homogeneous, and there is an involutive isometry of $M$ with at least one isolated fixed point, then $M$ is called a symmetric space.

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