Questions tagged [transversality]

In differential topology, transversality formalizes the idea of a generic intersection between two manifolds. It consists in asking an infinitesimal condition, namely on the tangent spaces, to be satisfied everywhere.

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Transversal invariant hyperbolas form manifold?

Recently I asked: Invariant curves... Let $\Bbb R^{1,1}$ be a semi-Riemannian manifold called Minkowski-space. Consider the invariant timelike hyperbolas of the lower branch (rectangular hyperbolas). $...
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If the invariant curves of two sub-manifolds are transversal, does this imply that the sub-manifolds are transversal?

Trying to determine the following: If the invariant curves of two sub-manifolds are transversal, does this imply that the sub-manifolds are transversal? An example I have of invariant curves, are ...
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Transversality and the hessian of critical ponint of a function

I am trying to do the following exercise: Let $f\in C(\mathbb{R^n,\mathbb{R}})$. Consider the gradient vector field $\nabla f=(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})$. ...
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Example of immersed submanifold that intersects itself non-transversally

Two submanifolds are said to intersect transversally if, along its intersection, the tangent spaces of the two submanifolds span the tangent space of the ambient space. Can a hypersurface immersed in $...
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Orientability of a submanifold which is a preimage of a submanifold

I am reading Milnor & Stacheff, Characteristic Classes, Chapter 18. There is a short review of smooth manifolds, and there is a following statement: Suppose $f:M\to N$ is a smooth map between ...
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self intersection of two Lorentzian manifolds also Lorentzian in this case?

Consider a manifold $\zeta:=\{\varphi_S\}\cap\{\varphi_T\}$ as the non-empty transversal intersection between two Lorentzian manifolds, s.t. all three manifolds have dimension $D=3+1.$ Is $\zeta$ also ...
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The Transverality Theorem in Differentiable Topology by Guillemin and Pollack

In Chapter 2, Section 3 of the book, most of the theorems requires the codomain $Y$ to be a manifold without the boundary and the submanifold $Z$ to be boundaryless as well. But I don't see why the ...
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29 views

Transversality of two mappings and diagonal

I've never taken differential topology and am confused by the definition of transversality, and while trying to solve the following I got stuck. Given smooth manifolds and maps $f:M\to N$ and $g:P\to ...
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Prove that the boundary orientation of $S^k = \partial B^{k+1}$ is the same as its preimage orientation

I would like to verify if my approach to this problem is the correct one or not. This problem is from "Differential topology" by Victor Guillemin and Allan Pollack . More specifically is the problem 3....
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existence of a transverse homotopic function - example

I would like to find an example for the following theorem: Let $f: M \to N$ be a smooth map. There exists a map arbitrarily close to $f$, and homotopic to it, which is transverse to $Z$. In the ...
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Intuition behind transversal intersection between manifolds

I am following the book "Differential Topology" by Allan Pollack and I am a little bit stuck with transversal intersection between manifolds. I understood the proofs presented in section 5 of such ...
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Existence of a transversal map prevents density?

Let $S$ be a $C^{\infty}$-submanifold of $N$ and suppose that $S-N$ is dense in $N$, where $M,N$ are $m$ and $n$ dimensional $C^{\infty}$-manifolds, respectively (without boundaries). In this post ...
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The Theory of Finite Groups - An introdution - Hans Kurzweil, Bernd Stellmacher.

Assume that $G$ allows a direct decomposition $$G = E_1 × ··· × E_n$$ that is invariant under A, i.e., $E_i^a \in {E_1,...,E_n}$ for all $a ∈ A$ and $i \in {1,...,n}.$ Under the additional hypothesis ...
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78 views

Transverse intersection is a well-defined operation on cobordism classes

Let $M$ be a smooth manifold of dimension $n$, $\textsf{FC}_p (M)$ the set of all codimension $p$ framed cobordism classes of $M$. Just as a technical note, we are only considering compact ...
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Diffeomorphic intersection of transverse submanifolds

Let $A,B$ be compact manifolds, $f:A\times[0,1]\to M$ and $g:B\times[0,1]\to M$ smooth maps such that for all $t\in [0,1]$, $f_t=f(\cdot,t)$ and $g_t=g(\cdot,t)$ are embeddings. Note $A_t=f_t(A)$ and $...
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Additivity of Codimension in Transversality

In Guillemin and Pollack's Introduction to Differential Topology, the following theorem is given: Theorem. The intersection of two transversal submanifolds $X,Z$ of $Y$ is again a submanifold. ...
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Uniform and Transversal Matroid

I'm reading Oxley's book Matroid Theory, and I read something that is not trivial for me... The book says that every uniform matroid is also transversal, but I don't understand why! I know that a ...
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Transversality Conditions in Optimal Control

I am trying to solve an optimal control problem with two state equations, $x_t$ and $y_t$, and fixed end-points, i.e., $t\in[0,T]$. The problem is standard except that the initial value, $x_0$ and ...
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Confusion regarding Transversality Condition

I am required to solve the following PDE using the Method of Characteristics: $$yu_x-xu_y=0$$ I was able to do that (hopefully successfully) and I got that $u(x,y)$ is given by: $$u(x,y)=F(x^2+y^2)$...
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Transversal vector field

Let $X$ be a vector field on some manifold $M$, $Y \subset M$ a submanifold. What does it mean for $X$ to be transversal to $Y$ , denoted $Y \pitchfork X$? It appears in my lecture in the following ...
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Minimal Transversal Proof when you have Axiom of Choice

I need help with one of my Set Theory homework tasks. This problem is in (ZFC) ( With Axiom of Choice). Can someone give me some idea with this proof. I thought that I can construct transversal and ...
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Co-orientation of subspaces of a vector space: $\mathcal{B} \sim \mathcal{B'} \iff \det_{(\mathcal{B},\mathcal{B_0})}(\mathcal{B'},\mathcal{B_0})>0$

This is question is regarding co-orientation of vector-space. We have following: I have worked with orientations of vector spaces and manifolds before, hence showing that sign of determinant of ...
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Confusion about transversality at a point

When reading this post, I become confused about what it means for a map to intersect transversally. To help conceptualize things better here's my question. Let $f:\mathbb{R}^d \rightarrow \mathbb{...
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Coordinates functions and Morse function.

I'm trying to resolve the following problem: Let $X$ be a submanifold of $\mathbb{R}^N$ of dimension $k$. Show that there exists $l : \mathbb{R}^N \rightarrow \mathbb{R}$ linear such that its ...
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Existence of a solution of a (quasi)-linear PDE of first order

Theorem 1: Consider the Cauchy-Problem $$ \begin{cases} a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)\\ \Gamma(s)=(x_0(s),y_0(s),u_0(s)) \end{cases} $$ Assume that there exists an $S_0\in\mathbb R$ for which ...
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Intersection of the image of a manifold and a line should be finite.

The problem is the following: Let $f: M^n \rightarrow \mathbb{R}^{2n+1}$ be a smooth map such that $0 \notin f(M)$. Show that there exists a line in $\mathbb{R}^{2n+1}$ such that $f(M) \cap L$ is a ...
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“Genericity” of immersions for maps $\mathbb{R^m} \rightarrow \mathbb{R^n}, n \geq 2m$.

I'm stucked in the following problem: Let $f: \mathbb{R^m} \rightarrow \mathbb{R^n}$ be a differentiable map with $n \geq 2m$. Let $\varepsilon >0$, prove that there exists $\alpha : \mathbb{R^m} \...
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Show that if $H$ is transitive on $(y_1,…,y_{k-1})$ then $\Gamma$ is $(s+1)$-arc-transitive.

Let $\Gamma$ be an $s$-arc-transitive graph that is regular of valency $k$. Let $A=(x_0,...,x_s)$ be an $s$-arc in $\Gamma$ and let $H$ be the stabilizer of the $s$-arc $A$. Let $y_1,...,y_{k-1}$ the ...
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Prove that $\textbf{B}$ is an $Aut(\Gamma)$-invariant partition.

Let $\Gamma$ be a vertex-transitive digraph with $\Delta \leq \Gamma $. Let $\textbf{C}$ be the collection of all subdigraphs of $\Gamma$ isomorphic to $\Delta$. Suppose that $\textbf{B} =({V(C):C \in ...
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What is the definition of transversality used in this context?

Now, I've been reading V.I. Arnol'd, Characteristic Class Entering in Quantization Conditions and I'm stuck on page 10. Here is the link He tolds that $uM^{(n)}$ is such that its tangential ...
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45 views

Transversality on manifolds

Let be $M^{n}$ a differentiable manifold and $N_{1}$,$N_{2}$ two submanifolds of $M$. We say that $N_{1}$ and $N_{2}$ intersect transversely if $N_{1} \cap N_{2} = \emptyset$ or for all $p \in N_{1} \...
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Reference request: clean intersection between submanifolds

Let $N_1,N_2$ be two smooth submanifolds in an ambient manifold $M$. There are two definitions of clean intersection between $N_1$ and $N_2$: $N_1$ and $N_2$ intersect cleanly if $N_1\cap N_2$ is a ...
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Viewing a closed subset in a Euclidean space as a submanifold in higher dimension

By Manifolds I will just mean smooth manifolds . If $C$ is any closed subset of $\Bbb R^k$, then show that $\exists$ a submanifold $X$ of $\Bbb R^{k+1}$ such that $X \cap \Bbb R^k = C$ . I here ...
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Transversality condition for infinite horizon from Karush-Kuhn-Tucker?

A few but influential papers in macroeconomics cite a working paper titled "The Kuhn-Tucker Theorem Implies the Transversality Condition at Infinity" by P. Romer (yes, that Paul Romer) and T. ...
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Singular points of projective hypersurface and Veronese map

Let $S = \{f = 0\} \subset \mathbb{P}^n$ be a hypersurface of degree $d$. Then $S$ is the intersection of $v(\mathbb{P}^n)$ with a unique hyperplane $H \subset \mathbb{P}^N$, where $v: \mathbb{P}^n \...
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117 views

Normal Bundle over Transverse Intersection

From Bott and Tu's Differential Forms in Algebraic Topology: Two submanifolds $R$ and $S$ in $M$ are said to intersect transversally iff $$T_xR + T_x S = T_x M$$ for all $x \in R \cap S$. For such ...
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Why do Guillemin and Pollack avoid the term “submanifold with boundary”?

On p. 60 of the book "Differential Topology" by named authors they state: Theorem. Let $f$ be a smooth map of a manifold $X$ with boundary onto a boundaryless manifold $Y$, and suppose that both $f:...
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Images of transverse maps intersect in a discrete set

I am studying transversality. There is a result that I would intuitively say is true (for example by locally identifying the manifold with its tangent space), but I can't get around the Differential ...
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transversal intersection of three curves in $\mathbb{R}^3$

Let me start by saying this is my first post. I am new to intersection theory and am currently reading Guillemin and Pollack's Differential Topology and have a question on the intersection of $3$ ...
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Transversality condition for an optimal control problem

Let the optimal control problem: $$\max \int_a^bF(t,x(t),u(t))dt$$ $$x'(t)=g(t,x(t),u(t)) \; for \; all \; t\in [a,b]$$ $$x(a)=A$$ $$x(b)\geq0.$$ Unising the maximum principle and the adjoint state $...
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Metric on a manifold with boundary, with prescribed orthogonal

Let $M$ be a compact manifold with boundary $\partial M$. Suppose that we are given a hyper surface $A$ which is transversal to $\partial M$. Is there a way to construct a metric $g$ on $M$ such ...
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Homotopy type of transversal submanifolds through deformation

Let $A,B \subset M$ be two transversal submanifolds of a compact manifold $M$. It seems rather intuitive that if $A$ and $B$ are deformed (say smoothly) in a way that they remain transversal to each ...
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71 views

Question about connected manifold

I need some tip to prove the following: If $N^{n}$ is a connected manifold and $M^{m}$ is a closed submanifold of $N$, such that $n-m\geq 2$, then $N-M$ is connected. I am supposed to use ...
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Suppose that $S_1$ and $S_2$ are smooth surfaces in $\mathbb{R}^n$

(a) Let $n = 3$ and suppose that they intersect at a point $p$ and do not have the same tangent plane at that point. Show that $p$ is not an isolated point of $S_1 \cap S_2$. (b) Let $n = 4$ and ...
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Transversality condition for infinite time horizon; Pontryagin Maximum principle

I want to solve a optimal control problem with the Pontryagin-Maximum-Principle, given following differential equation: $\dot x(t)=(a+b-c(t))x-c(t)\\ x(0)=x_0$ Regarding to the conditions: $x\geq0$ ...
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The diagonals $\Delta=\{(v,v)\mid v\in V\}$ is transversal to $W=\{v,Av\mid v\in V\}$ iff $+1$ is not an eigenvalue of A

Learning Differential topology, Sorry for asking anything trivial. I am stuck in this question: Let $V$ be a vector space and let $\Delta$ be the diagonal of $V \times V$ . For a linear map $A : V ...
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Consider the real-valued function $M:=\{(x,y,z)| (2 - (x^2 + y^2)^{1/2})^2 + z^2=1\}$ defined on $\mathbb{R}^3-\{(0, 0, z)\}$.

Show that the manifold $N=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2= 4\}$ is transverse to M. Identify the resulting manifold $N\cap M$. My Attempt: Pardon me for something vacuous, as I am a beginner in ...
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45 views

Transverality Condition $u_t+uu_x=-2u$

I am given the quasi-linear PDE, $$u_t+uu_x=-2u,$$ with the boundary condition $u(0,t)=e^{-t}$. I am trying to compute the transversality condition. I proceed as $$\text{det}(J)=\begin{vmatrix} \...
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269 views

Show that embeddings, diffeomorphisms, etc. are stable classes of maps

This is part of Problem 16 in Chapter 6 of Lee's Smooth Manifolds. Let $N,M,S$ be smooth manifolds. A smooth family of maps is a collection $\{F_s:N\to M \;|\; s\in S\}$ such that $F_s(x)=F(x,s)$ ...
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172 views

Definition of “transversal intersection” for piecewise linear submanifolds

I'm working with knots in the PL category. In "Surface Knots in 4-space" of Seiichi Kamada, the author states on p. 26 that the linking number of two oriented knots $K$ and $J$ in $S^3$ is the ...