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Is the standard smooth structure of $S^{7}$ the only structure for which each of the following canonical embedding is an smooth embedding?: 1)$S^{6}\to S^{7}$ 2)$S^{7}\to S^{8}$

Is there a manifold $M$ such that for every $x\in M$, $M-\{x\}$ is not parallelizable but there is a finite set $S\subset M$, with $\# S>1$, such that $M-S$ is parallelizable?

What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?: There is an $n$ dimensional sub vector space $V\su …

Let $M$ be a $n$ dimensional manifold and $p:TM\to M$ be the projection map. Then $\ker Dp$ is a $n$ dimensional vector bundle on $TM$, as a sub bundle of $TT(M)$. For what type of manifolds, …

Let $M$ be a (not necessarily compact)) smooth manifold. 1.Is there a smooth map $f:M\to \mathbb{R}$ and an open covering $\mathbb{R}=\cup U_{\alpha}$ such that each $f^{-1}(U_{\alpha})$ i …

Let $M$ be a manifold with the property that $f^{*}(TM)$ is isomorphic to TM, for every diffeomorphism $f$ on $M$. Does this imply that $M$ is parallelizable?

According to the interesting comment of Mohammad F Tehrani, I revise the question as follows: Assume $n>2$. For what type of compact n dimensional manifolds $M$ we can say: For every smooth em …

Let $(E,M,p)$ be a n dimensinal smooth vector bundle where $M$ is a k dimensional manifold. We assign to $M$, two different vector bundles $F_{1}$ and $F_{2}$ over $M$ as follows: 1)$TE$ is a vec …

Is there a manifold $M$ for which $\chi^{\infty} (M)$, the lie algebra of smooth vector fields on $M$ contains all finite dimensional Lie algebras(Up to isomorphism)? A weaker question: Is there a …

Is there an atlas $\mathcal{A}$ for $\mathbb{C}P^2 \setminus \{pt\}$ such that all transition maps of this atlas are affine maps?

Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ whose image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?

Let $M$ be a smooth n dimensional manifold. The foliation values of $M$, denoted by $F(M)$, is defined as \begin{equation} F(M)=\{ 1\leq k\leq n\mid \text{there exist an smooth $k$ dimensional fo …

In the following question, we defined the foliation values of an smooth manifold; Foliation values of a manifold Let $S_{i}$'s, $i\in \{0,1,\ldots,27\}$, be the smooth structures of topological $S^{ …

Is the unit tangent bundle of $S^{n}$ a parallelizable manifold. This is motivated by the fact that $TS^{n}$ is parallelizable?

Let $X$ be a Hausdorff topological space with the following property: For every continuous function $f:X\to X$, there is a finite subset $S\neq \emptyset$ of $X$ with $F(S)\subset S$ Do …