query: Definition of embedded submanifolds of $\mathbb{R}^n$.
positive_passages.docid: 148685
positive_passages.text: Equivalences of definitions of submanifolds in $\mathbb{R^n}$
negative_passages.docid: 148633
negative_passages.text: Description of varieties in $\mathbb{P}^2\times \mathbb{P}^1$
negative_passages.docid: 127886
negative_passages.text: Artin's lemma and some morphisms.
negative_passages.docid: 97637
negative_passages.text: Showing that isotropy groups are submanifolds
negative_passages.docid: 96567
negative_passages.text: Immersion of $\mathbb{R}$ to $\mathbb{R}^2$
negative_passages.docid: 133413
negative_passages.text: Proof of paracompactness of CW-complexes (J. Lee, Introduction to Topological Manifolds)
negative_passages.docid: 157742
negative_passages.text: Prove that $U(n)$ is a manifold
negative_passages.docid: 70576
negative_passages.text: Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
negative_passages.docid: 90092
negative_passages.text: Showing that $\bar{\mathbb{B}}^n$ is a manifold with boundary (Lee ITM Probelm 3-4)
negative_passages.docid: 78085
negative_passages.text: A non orientable closed surface cannot be embedded into $\mathbb{R}^3$
negative_passages.docid: 101860
negative_passages.text: Relating the normal bundle and trivial bundles of $S^n$ to the tautological and trivial line bundles of $\mathbb{R}P^n$
negative_passages.docid: 72548
negative_passages.text: Lie group embeddings $SO(10) \supset SU(3) \times SU(2) \times U(1)?$
negative_passages.docid: 122410
negative_passages.text: Can one build a homology theory using submanifolds and their boundaries?
negative_passages.docid: 29269
negative_passages.text: fundamental group of $GL^{+}_n(\mathbb{R})$
negative_passages.docid: 39094
negative_passages.text: Lie bracket of vector fields on $\Bbb R^{n}$
negative_passages.docid: 64920
negative_passages.text: Boundary Question in $\mathbb{R}^{2}$ (Manifolds)
negative_passages.docid: 150987
negative_passages.text: Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent