query: Why does no minimal surface in $\mathbb{R}^3$ exist that is diffeomorphic to the $2$-sphere?
positive_passages.docid: 142503
positive_passages.text: There are no compact minimal surfaces
negative_passages.docid: 93669
negative_passages.text: Is every embedded submanifold globally a level set?
negative_passages.docid: 147310
negative_passages.text: How does one prove that the Klein bottle cannot be embedded in $R^3$?
negative_passages.docid: 19280
negative_passages.text: Jordan Canonical Form determined by characteristic and minimal polynomials in dimension $3$, but not beyond
negative_passages.docid: 119579
negative_passages.text: $S^2 \times S^2$ is diffeomophic to $G_2(\mathbb{R}^4)$
negative_passages.docid: 84148
negative_passages.text: Why is $\mathbb{R}^2$ not a subspace of $\mathbb{R}^3$?
negative_passages.docid: 2348
negative_passages.text: Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic
negative_passages.docid: 108909
negative_passages.text: Is the unit sphere in $\Bbb R^4$ path connected?
negative_passages.docid: 35305
negative_passages.text: Why is $\mathbb{R}^2$ not a subset and /or a subspace of $\mathbb{R}^3$?
negative_passages.docid: 45187
negative_passages.text: Symplectic form on 2-sphere?
negative_passages.docid: 9793
negative_passages.text: Why is the image of a smooth embedding $f: N \to M$ an embedded submanifold?
negative_passages.docid: 92504
negative_passages.text: Criteria of a trivial G bundle of a 3-manifold
negative_passages.docid: 143269
negative_passages.text: Existence of area-preserving diffeomorphism between two sets in $\mathbb{R}^2$
negative_passages.docid: 133465
negative_passages.text: Stokes and Gauss' Divergence theorem on a closed smooth surface in $\Bbb R^3$
negative_passages.docid: 41872
negative_passages.text: Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$
negative_passages.docid: 110472
negative_passages.text: Existence of a Pivot Point in Euclidean Geometry
negative_passages.docid: 27746
negative_passages.text: Autohomeomorphism of the unit disk which is constant on the boundary, and which takes 0 to a prescribed point