# 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.

109 questions
Filter by
Sorted by
Tagged with
13 views

35 views

### 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 ...
35 views

### 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 ...
46 views

### 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 ...
29 views

55 views

### 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. ...
60 views

### 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 ...
34 views

### 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 ...
39 views

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: $$u(x,y)=F(x^2+y^2)... 0answers 59 views ### Transversal vector field Let X be a vector field on some manifold M, Y \subset M a submanifold. What does it mean for X to be transversal to Y , denoted Y \pitchfork X? It appears in my lecture in the following ... 1answer 116 views ### Minimal Transversal Proof when you have Axiom of Choice I need help with one of my Set Theory homework tasks. This problem is in (ZFC) ( With Axiom of Choice). Can someone give me some idea with this proof. I thought that I can construct transversal and ... 1answer 65 views ### Co-orientation of subspaces of a vector space: \mathcal{B} \sim \mathcal{B'} \iff \det_{(\mathcal{B},\mathcal{B_0})}(\mathcal{B'},\mathcal{B_0})>0 This is question is regarding co-orientation of vector-space. We have following: I have worked with orientations of vector spaces and manifolds before, hence showing that sign of determinant of ... 0answers 31 views ### Confusion about transversality at a point When reading this post, I become confused about what it means for a map to intersect transversally. To help conceptualize things better here's my question. Let f:\mathbb{R}^d \rightarrow \mathbb{... 0answers 47 views ### Coordinates functions and Morse function. I'm trying to resolve the following problem: Let X be a submanifold of \mathbb{R}^N of dimension k. Show that there exists l : \mathbb{R}^N \rightarrow \mathbb{R} linear such that its ... 0answers 13 views ### Existence of a solution of a (quasi)-linear PDE of first order Theorem 1: Consider the Cauchy-Problem$$ \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 ... 1answer 40 views ### 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 ... 1answer 36 views ### “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} \... 0answers 20 views ### 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 ... 0answers 17 views ### 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 ... 0answers 27 views ### 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 ... 1answer 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} \... 3answers 187 views ### 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 ... 1answer 55 views ### 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 ... 0answers 32 views ### 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. ... 1answer 78 views ### 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 \... 1answer 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$$for all x \in R \cap S. For such ... 0answers 45 views ### Why do Guillemin and Pollack avoid the term “submanifold with boundary”? On p. 60 of the book "Differential Topology" by named authors they state: Theorem. Let f be a smooth map of a manifold X with boundary onto a boundaryless manifold Y, and suppose that both f:... 0answers 23 views ### Images of transverse maps intersect in a discrete set I am studying transversality. There is a result that I would intuitively say is true (for example by locally identifying the manifold with its tangent space), but I can't get around the Differential ... 0answers 48 views ### transversal intersection of three curves in \mathbb{R}^3 Let me start by saying this is my first post. I am new to intersection theory and am currently reading Guillemin and Pollack's Differential Topology and have a question on the intersection of 3 ... 0answers 40 views ### Transversality condition for an optimal control problem Let the optimal control problem:$$\max \int_a^bF(t,x(t),u(t))dtx'(t)=g(t,x(t),u(t)) \; for \; all \; t\in [a,b]x(a)=Ax(b)\geq0.$$Unising the maximum principle and the adjoint state ... 0answers 38 views ### 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 ... 0answers 52 views ### 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 ... 2answers 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 ... 1answer 31 views ### 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 ... 0answers 85 views ### 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 ... 1answer 132 views ### 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 ... 1answer 18 views ### 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 ... 0answers 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} \...
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 $F_s(x)=F(x,s)$ ...
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 ...