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 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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引入完全非平行的沉浸

一个光滑的 $n$ 维流形 $M \to \mathbb{R}^q$ 的浸入被称为完全非平行的，如果对于每个不同的 $x, y \in M$，$f(x) 处的切空间$ 和 $f(y)$ 不包含平行线。给定一个流形 $M$，我们寻求最小维度 $TN(M)$，使得存在一个完全非平行的浸入 $M\to\mathbb{R}^{TN(M)}$。与 Ghomi 和 Tabachnikov 研究的完全倾斜嵌入类似，我们发现完全非平行浸入与广义向量场问题、实射影空间的浸入问题和非奇异对称双线性映射有关。我们对完全非平行浸入的研究遵循了最近研究条件的趋势，这些条件表现在 $M$ 不同点的 k 元组的配置空间 $F_k(M)$ 上；例如，k-regular embeddings、k-skew embeddings，k-邻域嵌入等。 通常，满足这些配置空间条件之一的映射会在配置空间 $F_k(M)$（或其束）上引入一些 $S_k$-等变映射，并且可以使用 Stiefel 计算障碍物- 惠特尼课程。然而，这种条件的存在问题相对未得到研究。我们的主要结果是惠特尼型定理：每个流形 $M$ 都承认完全不平行地浸入 $\mathbb{R}^{4n-1}$，比泛型给定的维度少一个维度。我们首先研究局部问题，这需要对非奇异对称双线性映射的空间有透彻的了解，然后使用 Gromov 和 Eliashberg 的去除奇异点 h 原理技术建立主要定理。结合最近的戴维斯非浸入式定理，