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My question is, in order to completely model our universe's (3+1)-dimensional spacetime as a manifold in a higher-dimension, flat spacetime, how many dimensions are needed? Will a finite number of dimensions suffice, or are an infinite number of flat dimensions required to completely embed all solutions to the Einstein Equation?Our curved spacetime need not be embedded in some higher-dimensional flat spacetime for us to understand its curvature, or the concept of tangent vector. The mathematics of tensor calculus is designed to let us handle these concepts `intrinsically' i.e., working solely within the 4-dimensional spacetime in which we find ourselves.

If anyone can refer me to texts or publications that treat GR in this manner (as a curved manifold in higher-d flat spacetime) it would be much appreciated.