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Line 7: ==Nash–Kuiper theorem ({{math\|''C''<sup>1</sup>}} embedding theorem) {{anchor\|Nash–Kuiper theorem}}== Given an {{~~math}"~~mvar\|m"}}-dimensional Riemannian manifold {{math\|(''M'', ''g'')}}, an ''isometric embeddding'' is a continuously differentiable [[topological embedding]] {{math\|''f'': ''M'' → ℝ<sup>''n''</sup>}} such that the [[pullback]] of the Euclidean metric equals {{mvar\|g}}. In analytical terms, this may be viewed (relative to a smooth [[coordinate chart]] {{mvar\|x}}) as a system of {{math\|{{sfrac\|1\|2}}''m''(''m'' + 1)}} many first-order [[partial differential equation]]s for {{mvar\|n}} unknown (real-valued) functions: :<math>g_{ij}(x)=\sum_{\alpha=1}^n\frac{\partial f^\alpha}{\partial x^i}\frac{\partial f^\alpha}{\partial x^j}.</math> If {{mvar\|n}} is less than {{math\|{{sfrac\|1\|2}}''m''(''m'' + 1)}}, then there are more equations than unknowns. From this perspective, the existence of isometric embeddings given by the following theorem is considered surprising.

Nash embedding theorems: Difference between revisions