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Questions tagged [differential-topology]

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. Somewhat simplified view of this field is that it describes the setting to which the notion of differentiable function can be generalized from the more familiar case of functions $\mathbb R^n\to\mathbb R^k$. Differential topology is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Calculate $\deg(f)$

According to Guillemin and Pollack, Differential Topology Page 109, $f: X \to Y$ are appropriate for intersection theory ($X,Y$ are boundaryless oriented manifolds, $X$ is compact), when $Y$ is connected and has the same dimension as $X$, we define…
1LiterTears
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What does the notation $\epsilon(f(x))s$ mean?

I am very, very confused with the notion $\epsilon(f(x))s$. To my understanding, $s$ is a map sends to $F(x,s)$, and $\epsilon$ is the distance function given a point $f(x)$. So what does $\epsilon(f(x))s$ means, when we just put them together?…
1LiterTears
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Projection, canonical immersion/submersion - are they equivalent, and are they open maps?

I am very confused with the concept of projection with the introduction of immersion and submersion. By local immersion/submersion theorem, for a simmersion/submersion $f$, there is is a canonical immersion/submersion locally is equal to $f$. So…
WishingFish
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local diffeomorphism on $\mathbb{R}$ and on manifolds.

I find the proof of diffoemorphism in Guillemin & Pallock's Differential Topology 1.3.3 is more or less independent of the fact that the manifold happen to be $\mathbb{R}$, and therefore are the same. Then I am asking if my two proofs (primarily the…
WishingFish
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Show that $\pi: \mathbb S^{2n+1} \to \mathbb {CP}^{n}$ is a bundle of fibre $\mathbb S^1$?

Show that $\pi: \mathbb S^{2n+1} \to \mathbb {CP}^{n}$ is a bundle of fibre $\mathbb S^1$ ? My attempt : Let $U_i=\{[z]\,, z_i>0\}$ and $O_i=\{ z\in \mathbb S^{2n+1}: z_i>0\}$. We have that $U_i=\pi(O_i)$ and $\mathbb{CP}=\cup_{i=1}^{n+1} U_i$ and…
H_K
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The general idea of prove openness.

I never really get the idea of proofs involves openness, here's an example: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a local diffeomorphism. Prove that the image of $f$ is an open interval. So, is the general principle is to show: Step n:…
WishingFish
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Immersion is a diffeomorphism

Suppose $X$ is a smooth, compact, connected $n$-manifold without boundary which admits an immersion to $S^n$. Show that if $n>1$, then this immersion is a diffeomorphism. Thanks for the very inspiring mentors, here I got some thoughts $df_x$ is…
1LiterTears
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Homomorphisms of Lie Groups

$SO(3)$ denotes the special orthogonal group, which is the (open) subset of $O(3)$ on which the determinant is one. I have shown that every element of $SO(3)$ fixes a line in $\mathbb{R}$ pointwise and then $SO(3)$ is connected. Also, suppose…
1LiterTears
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Find an closed 1-form on $\mathbb{R}^2 \backslash (0,0)$ that is not exact.

I need help with the following problem. I am not sure how start and I would be very appreciative if someone could help me with this (I believe easy?) example. Find an closed 1-form on $\mathbb{R}^2 \backslash (0,0)$ that is not exact. Thanks in…
gary
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Are there $\pi$ -Dimension?

Are there $\pi$ -Dimension ? When I was thinking of dimensions such as about n-balls. I asked myself why isn't there a $\pi$-Ball. we always talk about n being a natural number. I know the illustration is difficult but the same applies for the 4th…
Maths
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explicit formula for embedding projective spaces into euclidean space

i can't seem to find very many good answers for this. most of the theorems out there use cohomology methods (Stiefel-Whitney classes, etc.) for proving that projective spaces can be embedded into Euclidean spaces of certain dimensions. Don Davis has…
Kevin Knudson
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Showing a hypersurface is contained in a level set of a regular value

I'm stuck on the following problem: let $S$ be a compact orientable hypersurface in the symplectic manifold $(M,\omega)$. Prove that there exists a smooth function $H: M \to \mathbb R$ such that $0$ is a regular value of $H$ and $S \subset…
Eric O. Korman
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Differentiable map sending square to circle

The question is that simple. Is there any known consruction of a differentiable map $\phi \colon U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that sends a closed square $Q\subset U$ to a circle? Of course, the derivative of $\phi$ at the…
Alan Muniz
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Poincare lemma or its converse?

In the old days the name "Poincare lemma" used to be the statement that $d^2=0$. This is the usage of books like Flanders' "Differential Forms with Applications to the Physical Sciences ", Bishop and Goldberg's "Tensor Analysis on Manifolds" and…
mike stone
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Hyperbolic equilibrium point on the sphere

I would like to construct an explitic smooth vector field on the sphere such that the north and south pole are hyperbolic equilibrium points. Mi idea is this To construct a vector field in $R^2$ with the origin as the hyperbolic point. Say $V: R^2…
Math Guy
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