I know these manifolds can be expressed as constraint equations. For example if we consider a sphere embedded in a 3D Euclidean space, the sphere can actually be expressed in a set of intrinsic coordinates restricted to the manifold.For example we can use polar coordinates. But what I am wondering is these are just 2 dimensional and 3 dimensional. Are there any specific examples for higher dimensional embeddings?
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The $n$-sphere $S^n={ (x_1, x_2, \cdots, x_n) |\sum_{i\leq n} x_i^2=1}$ can be shown to be a smooth manifold endowed with coordinates given by constraint equations. In general, a powerful tool for showing that the set of solutions to some constraint equation is a smooth manifold in a natural way is done using the "Regular value theorem". – J.V.Gaiter Nov 22 '21 at 23:56
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@J.V.Gaiter Have you got any references? – Jasmine Nov 23 '21 at 00:13
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This is covered in Lee's book on smooth manifolds in the chapter on submanifolds. – J.V.Gaiter Nov 23 '21 at 19:24