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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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Find the critical points of this function

Let $M={(x,y,z,w)∈R^4|x^4+y^4 + z^2 + w^2 = 1}$and let $f:M \rightarrow R$ be given by$f(x,y,z,w)=x^3 - z.$ a) Show that M is a manifold. b) Find the critical points of f. Part a is easy but how do you do part b?
Eddie
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Applications and Interpretation of Jacobian in Square-to-Torus and Square-to-Sphere Transformations

I am interested in the topological transformations that map a square to other surfaces, such as a torus and a sphere, particularly in the context of taking the Jacobian of these transformations. For the square-to-torus transformation, the…
Dwip Dalal
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Continuous pointwise orientation

Lee defines an orientation on a smooth manifold $M$ as a continuous pointwise orientation. A pointwise orientation (that is, a choice of orientation on each tangent space of $M$) is said to be continuous if every point of $M$ is contained in the…
warzasch
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Bundle morphisms in terms of sections

Let $E$ and $F$ be complex vector bundles on a smooth manifold $M$. I'm trying to prove that the $C^{\infty}(M)-$module morphisms $L: \Gamma(E) \rightarrow \Gamma(F)$ are in correspondence with the bundle morphisms $\tilde{L}: E \rightarrow F$. But…
Lazarus Frost
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Examples of Topological Spaces which are homeomorphic but not diffeomorphic.

I understand that two topological spaces being homeomorphic is, in some sense, a weaker form of isomorphism than the two spaces being diffeomorphic because diffeomorphism preserves more structure. What would be some simple examples of spaces which…
Keshav Balwant Deoskar
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Question about extending "locally" local sections

I wonder if the following is true: Let $\pi : E \rightarrow M$ be a vector bundle over a smooth manifold $M$, $U$ an open set of $M$, and $r \in \Gamma(U,E)$ a local smooth section defined on $U$. Then, for every point $p \in U$, there exists a…
Lazarus Frost
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Inmersion and submersion that is not a local diffeomorphism

I know that the maps between manifolds without boundary which are both a submersion and an immersion are exactly the local diffeomorphisms, and I wonder if it can happen that a map from a manifold without boundary to a manifold with boundary that is…
Lazarus Frost
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Examples of non-parametrizable sets?

Encountered the term parametrizable for the first time: The support of $\omega$ is contained inside a single parametrizable open subset $W$ of $X$. So I am just curious, what kind of sets are not parametrizable? The intersection of $\mathbb{R}^2$…
1LiterTears
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$f^*\left(\sum_I a_I \, dx_I\right) = \sum_I (f^*a_I) \, df_I$

By linearity of $f^*$, $$f^*\left(\sum_I a_I \, dx_I\right) = \sum_I f^* (a_I \, dx_I)$$ And if I want $df_I$, I would have to use the formula $f^*dx_i = df_i$. So $f^*$ disappears when introduced $df_i$ according to my…
1LiterTears
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Page of Milnor I don't understand

At page 22, when proving the "Homotopy lemma", Milnor uses the fact that $|f^{-1}(y) + g^{-1}(y)| =|f^{-1}(y)| + |g^{-1}(y)|$, (thus he concludes that $0\equiv \partial F = |f^{-1}(y)| + |g^{-1}(y)|$, i.e. the thesis). I don't see why this should be…
Stefano Saviani
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Graph of a smooth map is a smooth manifold but is it orientabl

I got to know that the Tangent Bundle of a smooth manifold is Orientable I read that the graph of a smooth map is a smooth manifold. Now I am thinking if it is always orientable or not. I think it is not. Define some map $f:U \to \mathbb{R}$ such…
permutation_matrix
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Intuition about manifolds with/without boundary

I saw that a $m-$manifold with boundary is defined by the following. Every point of $M$ has a neighborhood homomorphic to either an open ball in $\mathbb{R}^m$ or the upper half of $\mathbb{R}^m$ which is known as $\mathbb{H}^m$. But how do I…
permutation_matrix
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Ehresmann's lemma/ Ehresmann's Fibration Theorem Reference

I'm required to use Ehresmann's Lemma in a project to prove that something is a locally trivial fibration but I'm struggling to find a good reference for the theorem - i.e. it's stated nicely on Wikipedia etc but I can't find it in a book written in…
Hope Duncan
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Mayer Vietor Sequence for DeRham Cohomology

I am trying to understand the mayer-vietoris sequence for DeRham Cohomology. Assume $M=U \cup V$ and we want to show that the sequence $$ 0 \to \Omega(M) \to \Omega(U) \oplus \Omega(V) \to \Omega(U \cap V) \to 0$$ is a exact. Here I am bit unsure…
permutation_matrix
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Embedding from $\mathbb{RP}^{n}$ into $\mathbb{RP}^{m}$

Let $n \leq k$ positive integers show that the natural embedding $\mathbb{R}^{n+1} \hookrightarrow \mathbb{R}^{m+1}$ induces an embedding $\mathbb{RP}^{n} \hookrightarrow \mathbb{RP}^{m}$ This is my atemp: Let $j$ the embedding from…
Nick_W
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