# Tagged Questions

18 views

### Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
86 views

### How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
44 views

36 views

### About $C^{0}$ being topological manifold

Is that the reason why $C^{0}$ being topological manifold due to that $C^{0}=\phi$ which contains nothing? Correct me if I am wrong. I am new to differential topology.
39 views

### Why is the irrational winding of the torus not locally path connected?

The irrational winding of the torus given by the map $f\colon\mathbb{R}\to T^2$ where $f(t)=(e^{it},e^{i\alpha t})$ for some irrational $\alpha$. Wikipedia mentions this is not a regular submanifold, ...
85 views

### Is $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ diffeomorphic to $S^2$?

I was working on Problem 5-1 of Smooth Manifolds by Professor John Lee, and it lead me to wanting to show that $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ is diffeomorphic to $S^2$, and that is ...
38 views

### Smooth embeddings of the $2$-sphere

I have a past qual question here: given a smooth embedding $f \colon S^2 \to \mathbb{R}^3$, show that there must exist distinct points $p,q \in S^2$ such that the tangent planes to the embedded sphere ...
27 views

### Inward and outward pointing tangent vectors?

If $M^n$ is a smooth manifold with boundary and $p\in\partial M$, then $T_pM$ is the disjoint union of inward and outward point vectors, and $T_p\partial M$. If $(U,(x^i))$ is a smooth boundary chart ...
53 views

### How does $v\Phi^1=\cdots=v\Phi^k=0$ imply $v\in\ker d\Phi_p$?

I'm confused about an immediate corollary in John Lee's Smooth Manifolds. Proposition 5.38 says Suppose $M$ is a smooth manifold and $S\subset M$ is an embedded submanifold. If $\Phi\colon U\to N$ ...
39 views

81 views

### Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
60 views

### Derivative of the inclusion of a submanifold

I know there are other questions similar to this one, but I just want you to tell me if what I'm doing is rigth and how to improve it. The problem is the following: (I'm using the definitions by ...
33 views

### How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
134 views

### Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
85 views

### Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
125 views

### Product neighborhood theorem with boundary

The Product Neighborhood Theorem states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization of the normal bundle (wrt. some smooth ...
110 views

### Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
43 views

### Help understanding a proof in differential geometry

I was reading John Milnor's Topology from the Differentiable Viewpoint and there's a proof of the fundamental theorem of algebra at the end of the first chapter that I don't fully understand. I can ...
### What is $T^*(\mathbb{A}^1)$?
Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
### Interpretation of $p$-forms
Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...