# Questions tagged [smooth-manifolds]

For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.

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### Local structure of symplectic manifold

Algebraic geometry, in a naive setting, could be described as the study of spaces that locally are the solutions of systems of polynomial equations. Similarly, locally any smooth manifolds can be ...
1 vote
50 views

### Question about the definition of a vector field on a manifold

I am reading do Carmo's Riemannian Geometry, and he defines the tangent bundle $TM$ of a differentiable manifold $M$ as $\{ (p, v) : p \in M, v \in T_pM\}.$ That's fine. Next, he defines (on page 25): ...
1 vote
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### Proposition 1.1.14 D.J Saunders on Bundle

everyone. I am studying D.J Saunders's book The Geometry of Jet Bundles. On proposition 1.1.14, a proof is given that the structure of the total space $E$ of a bundle $(E,\pi,M)$ depends on those of ...
22 views

### questions about regular level set theorem and manifolds

Regular level set theorem says that every regular level set of a smooth map is a closed embedded submanifold whose codimension is equal to the dimension of the range. Is it if and only if? For example ...
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### Show that the product of any number of spheres can be embedded in some Euclidean space with codimension one

This problem is already solved in an elegant way here: Product of spheres embeds in Euclidean space of 1 dimension higher But I was trying to use a different approach: I'm using induction, it's clear ...
1 vote
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1 vote
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### What does "mod" mean in the context of this tangent space proof in Warner?

I'm trying to read Frank Warner's Foundations of Differentiable Manifolds and Lie Groups and got confused with Theorem $1.17$ as some who does not have a pure mathematics background. Let $F_m$ be the ...
1 vote
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1 vote
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### Manifolds - Inverse Function Theorem Form?

For context, I am reading Appendix F is Milnor's book on Sullivan's No Wandering Theorem and this is where he uses the following manifolds result: Let $F:X \rightarrow Y$ be a smooth function between ...
105 views

### Quotient of cohomology groups with different coefficents

Let $M$ be a smooth manifold, (if necessary I'm ok with assuming that $M$ is four dimensional, orientable, and closed). I wish to understand the quotient: $$Q=H^1(M;\mathbb{R})/H^1(M;\mathbb{Z})$$ I ...
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### Proving that the atlas of an open submanifold is smooth

$\def\AAA{\mathcal{A}} \def\sbe{\subseteq} \def\y{\psi} \def\w{\omega} \def\x{\chi}$ In section $1.26$ of Lee's Intro to Smooth Manifolds the concept of an open submanifold is introduced, but it is ...
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### Properly interpreting propositions 2.5 and 2.6 in Lee's Introduction to Smooth Manifolds

I want to make sure that I am correctly interpreting propositions $2.5$ and $2.6$ (picture below) of Lee's Introduction to Smooth Manifolds. Are the two following interpretations correct? In $2.5$ (b)...
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### Why is this subset associated to a $2$-tensor open and dense?

Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the ...
1 vote
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### Proving the image of a function of rank one is a curve using constant rank theorem

My question is based on this one and is prompted by my attempt to understand the constant rank theorem. Specifically, suppose I have a smooth map $F : M \rightarrow R^k$ where $M \subset R^n$ is an $m$...
### Why is $R_p:T_pM\times T_pM \to \operatorname{Hom}(E_p,E_p)?$
Let $M$ be a smooth manifold and $E\to M$ a vector bundle over $M$ and $\nabla:\Gamma(E) \to \Gamma(T^*M\otimes E)$ a connection on $E$. The curvature $R$ of the connection is defined as R: \...
### Are there "differentiable manifolds" that don't admit a $C^1$-structure
It is well known that every $C^1$ manifold admits a smooth manifold structure. What if we relax the definition of smooth manifold so the transition maps need only be differentiable? Does every such &...