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Totally skew embeddings of manifolds
"... Abstract. We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an ndimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generaliz ..."
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Cited by 16 (9 self)
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Abstract. We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an ndimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of nonsingular bilinear maps, and the immersion problem for real projective spaces. We use these connections and other methods to obtain several specific and general bounds for the desired dimension. 1.
Existence and nonexistence of skew branes
"... In mid1960s, H. Steinhaus conjectured that every closed smooth curve in ..."
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In mid1960s, H. Steinhaus conjectured that every closed smooth curve in
Central crosssections make surfaces of revolution quadric
 Amer. Math. Monthly
"... Abstract. When all planes transverse and nearly perpendicular to the axis of a surface of revolution S intersect it in loops having central symmetry, S must be quadric. Dedicated to Sue Swartz 1. ..."
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Abstract. When all planes transverse and nearly perpendicular to the axis of a surface of revolution S intersect it in loops having central symmetry, S must be quadric. Dedicated to Sue Swartz 1.
TOPOLOGY OF SURFACES WITH CONNECTED SHADES
"... Abstract. We prove that any closed orientable surface may be smoothly embedded in Euclidean 3space so that when it is illuminated by parallel rays from any direction the shade cast on the surface is connected. 1. ..."
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Abstract. We prove that any closed orientable surface may be smoothly embedded in Euclidean 3space so that when it is illuminated by parallel rays from any direction the shade cast on the surface is connected. 1.
SURFACES WITH CENTRAL CONVEX CROSSSECTIONS
, 904
"... Abstract. Say that a surface in S ⊂ R3 has the central plane oval property, or cpo, if • S meets some affine plane transversally along an oval, and • Every such transverse plane oval on S has central symmetry. We show that a complete, connected C2 surface with cpo must either be a cylinder over a ce ..."
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Abstract. Say that a surface in S ⊂ R3 has the central plane oval property, or cpo, if • S meets some affine plane transversally along an oval, and • Every such transverse plane oval on S has central symmetry. We show that a complete, connected C2 surface with cpo must either be a cylinder over a central oval, or else quadric. We apply this to deduce that a complete C2 surface containing a transverse plane oval but no skewloop, must be cylindrical or quadric. 1. Introduction and
Lemma 1.1 One has:
"... Abstract. We prove nonexistence of C 2smooth embeddings of ndimensional discs to R 2n such that the tangent spaces at distinct points are pairwise disjoint. ..."
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Abstract. We prove nonexistence of C 2smooth embeddings of ndimensional discs to R 2n such that the tangent spaces at distinct points are pairwise disjoint.
Nonexistence of ndimensional Tembedded discs in R 2n
, 2005
"... A number of recent papers concerned various nondegeneracy conditions on embedding and immersions of smooth manifolds in affine and projective spaces defined in terms of mutual positions of the tangent spaces at distinct points, see [1, 2, 3, 4, 6, 7, 9]. Following Ghomi [1], a C 1embedded manifold ..."
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A number of recent papers concerned various nondegeneracy conditions on embedding and immersions of smooth manifolds in affine and projective spaces defined in terms of mutual positions of the tangent spaces at distinct points, see [1, 2, 3, 4, 6, 7, 9]. Following Ghomi [1], a C 1embedded manifold M n ⊂ R N is called Tembedded if the tangent spaces to M at distinct points do not intersect. For example, the cubic curve (x, x 2, x 3) is a Tembedding of R to R 3, and the direct product of such curves gives a Tembedding of R n to R 3n. A Tembedding M n → R N induces a topological embedding of the tangent bundle TM → R N, hence N ≥ 2n. One of the results in [1] is that no closed manifold M n admits Tembeddings to R 2n. In this note we strengthen this result as follows. Theorem 1 There exist no C 2smooth Tembedded discs D n in R 2n. Proof. Arguing by contradiction, assume that such a disc D n exists. Choose the tangent space at the origin and its orthogonal complement as coordinate ndimensional spaces. Making D smaller, if necessary, assume that the disc is the graph of a (germ of a) C 2 smooth map f: R n → R n. Let U ⊂ R n be the domain of f. Partially supported by NSF 1 Let z = (u, f(u)) ∈ D where u ∈ U. The tangent space TzD is given by a linear equation y = A(u)x − b(u) where A(u) is an n × n matrix and b(u) is a vector in R n, both depending on u. In terms of f, they have the following expressions. Let f1,..., fn be the components of f. Lemma 1.1 One has: Aij = ∂fi, bi =