# Continuous map from one manifold to another

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## Main Question or Discussion Point

I need a reminder. What numbers or functions characterize a map from one manifold to another? More specifically, is there a continuous function that goes from one manifold to another to another to another is some parameterized way? What is that called?

I'm thinking of a manifold of spacetime. And with acceleration one goes from the initial frame of reference (the first manifold) to a frame of reference traveling with respect to the first frame. This can be viewed as a different manifold but shares some points with the first. Can this acceleration from one frame (manifold) to another frame (manifold) be characterized by numbers that tell us something about the mass or energy need to go from one frame to another? I hope this is enough information to generate answers. Thanks you.

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stevendaryl
Staff Emeritus
Now, you can think of a "coordinate system" as a map between manifolds: Spacetime is one manifold, $\mathcal{M}$ and $R^4$ is another manifold (the set of all possible 4-tuples $(x,y,z,t)$ of real numbers). A coordinate system $\mathcal{C}$ is a map from $\mathcal{M}$ to $R^4$ with the restriction that the map must be continuous and must have an inverse. (More generally, a coordinate system might be just a map from a region in $\mathcal{M}$ to $R^4$, rather than from all of $\mathcal{M}$.) A coordinate change can be thought of as a map between two different copies of $R^4$.