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  "body": "\n## Summary\n\nA submanifold of a [[smooth manifold|F45VQ47GPV]] $M$ is a subset that is itself a smooth manifold in a way compatible with $M$.\nThe most common kind, an embedded submanifold, looks near each of its points like a flat coordinate subspace $\\R^k \\subseteq \\R^n$.\nLevel sets of submersions, images of embeddings and graphs of smooth maps are submanifolds.\nA wider notion, the immersed submanifold, is the image of an injective [[immersion|6RW68JGBHC]] with its own topology.\n\n## Explanations\n\n### L1 human\n\nA submanifold is a smooth curve, surface, or higher-dimensional piece sitting inside a bigger smooth space.\nSeen through a good magnifying glass, it looks like a flat line or plane inside flat space.\nThe unit circle in the plane and the unit sphere in $\\R^3$ are submanifolds.\nThe union of the two coordinate axes in $\\R^2$ is not, because at the origin it looks like a cross, not like a line.\n",
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