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2008-05-18T20:10:16Z

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Vipul at 20:47, 13 January 2008

2008-01-13T20:47:33Z

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==Definition==

Let <math>M</math> be a [[differential manifold]]. A '''submanifold''' of <math>M</math> can be viewed as the following data: An abstract [[differential manifold]] <math>N</math> along with a [[smooth map]] <math>f</math> from <math>N</math> to <math>M</math> such that:

# The map <math>f</math> is an [[immersion]]; in other words, the induced map <math>(Df)_p</math> on the tangent space at any point <math>p \in M</math> is injective
# The map is injective i.e. <math>f(p) = f(q) \implies p = q</math>
# The map is a homeomorphism to its image

Note that when <math>N</math> is compact, the third condition is redundant, because [[tps:Injection from compact to Hausdorff implies embedding|any injective map from a compact space to a Hausdorff space is an embedding]].

To complete the definition, we need to observe that a submanifold is completely determined, upto diffeomorphism, by its set-theoretic image in the manifold. In other words, if <math>f_1:N_1 \to M</matH> and <math>f_2:N_2 \to M</math> are submanifolds with the same set-theoretic image, then there is a diffeomorphism <math>g:N_1 \to N_2</math> such that <math>f_1 = f_2 \circ g</math>.

Submanifold (differential sense) - Revision history

Vipul: 1 revision

Vipul at 20:47, 13 January 2008