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Introducing Totally Nonparallel Immersions
Geometric Topology · Submitted 25 Jul 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our main result is a Whitney-type theorem: every $n$-manifold $M$ admits a totally nonparallel immersion into $\mathbb{R}^{4n-1}$, one dimension less than given by genericity.
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The abstract, sentence by sentence
An immersion of a smooth $n$-dimensional manifold $M \to \mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y \in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion $M \to \mathbb{R}^{TN(M)}$. In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion problem for real projective spaces, and nonsingular symmetric bilinear maps. Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space $F_k(M)$ of k-tuples of distinct points of $M$; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, etc. Typically, a map satisfying one of these configuration space conditions induces some $S_k$-equivariant map on the configuration space $F_k(M)$ (or on a bundle thereof) and obstructions can be computed using Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied. Our main result is a Whitney-type theorem: every $n$-manifold $M$ admits a totally nonparallel immersion into $\mathbb{R}^{4n-1}$, one dimension less than given by genericity. We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. When combined with a recent non-immersion theorem of Davis, we obtain the exact value $TN(\mathbb{R}P^n) = 4n-1$ when $n$ is a power of 2. This is the first optimal-dimension result for any closed manifold $M \neq S^1$, for any of the recently-studied configuration space conditions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An immersion of a smooth $n$-dimensional manifold $M \to \mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y \in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. | background 0.98 | 0.09 | 0.02 | asserted 0.99 |
| s1 | Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion $M \to \mathbb{R}^{TN(M)}$. | background 0.81 | 0.25 | 0.03 | asserted 0.67 |
| s2 | In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion problem for real projective spaces, and nonsingular symmetric bilinear maps. | background 0.59 | 0.84 | 0.02 | asserted 0.99 |
| s3 | Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space $F_k(M)$ of k-tuples of distinct points of $M$; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, etc. | background 0.96 | 0.12 | 0.01 | asserted 0.99 |
| s4 | Typically, a map satisfying one of these configuration space conditions induces some $S_k$-equivariant map on the configuration space $F_k(M)$ (or on a bundle thereof) and obstructions can be computed using Stiefel-Whitney classes. | background 0.91 | 0.08 | 0.01 | asserted 0.82 |
| s5 | However, the existence problem for such conditions is relatively unstudied. | gap 0.99 | 0.10 | 0.01 | asserted 0.98 |
| s6 | Our main result is a Whitney-type theorem: every $n$-manifold $M$ admits a totally nonparallel immersion into $\mathbb{R}^{4n-1}$, one dimension less than given by genericity. | result 0.99 | 0.95 | 0.98 | asserted 1.00 |
| s7 | We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. | method 0.98 | 0.79 | 0.03 | asserted 0.99 |
| s8 | When combined with a recent non-immersion theorem of Davis, we obtain the exact value $TN(\mathbb{R}P^n) = 4n-1$ when $n$ is a power of 2. | result 0.98 | 0.78 | 0.05 | asserted 1.00 |
| s9 | This is the first optimal-dimension result for any closed manifold $M \neq S^1$, for any of the recently-studied configuration space conditions. | interpretation 0.83 | 0.82 | 0.11 | asserted 1.00 |
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Original abstract
An immersion of a smooth $n$-dimensional manifold $M \to \mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y \in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion $M \to \mathbb{R}^{TN(M)}$. In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion problem for real projective spaces, and nonsingular symmetric bilinear maps. Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space $F_k(M)$ of k-tuples of distinct points of $M$; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, etc. Typically, a map satisfying one of these configuration space conditions induces some $S_k$-equivariant map on the configuration space $F_k(M)$ (or on a bundle thereof) and obstructions can be computed using Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied. Our main result is a Whitney-type theorem: every $n$-manifold $M$ admits a totally nonparallel immersion into $\mathbb{R}^{4n-1}$, one dimension less than given by genericity. We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. When combined with a recent non-immersion theorem of Davis, we obtain the exact value $TN(\mathbb{R}P^n) = 4n-1$ when $n$ is a power of 2. This is the first optimal-dimension result for any closed manifold $M \neq S^1$, for any of the recently-studied configuration space conditions.
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Nonsingular (symmetric) bilinear maps have been studied, in part, due to their relationships to immersions and embeddings of projective spaces and to totally nonparallel immersions; see [ref] .
Spaces of embeddings: Nonsingular bilinear maps, chirality, and their generalizations · Nonsingular bilinear maps · 0.95
What it cites · 2 sentences without a confident label
Cites as background 1 paper
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* $k$-regular if for each $x ∈F_k(M)$, the collection of vectors ${f(x_1), …, f(x_k) }$ is linearly independent ( [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] , [ref] ). * $k$-skew if for each $x ∈F_k(M)$, the collection of affine subspaces ${T_f(x_1)^q, …, T_f(x_k)^q}$ of $^q$ is affinely independent ( [ref] , [ref] , [ref] , [ref] , [ref] ). * $k$-neighborly if for each $x ∈F_k(M)$, there is an affine hyperplane $H$ supporting $f(M)$ which touches $f(M)$ at exactly the points $f(x_1), …, f(x_k)$; that is, $H$ contains the points $f(x_1), …, f(x_k)$ and one of the open half-spaces determined by $H$ contains the remaining points of $f(M)$ ( [ref] , [ref] ).
Constructions of k-regular maps using finite local schemes · Introduction and Main Results · 0.98
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