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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Regarding Thom's Jet Transversality Theorem proof from Golubitsky-Guillemin (how the Parametric Transversality Theorem is applied)

I am trying to follow the Thom's Transversality Theorem proof from Golubitsky-Guillemin "Stable mappings and Their Singularities" (theorem 4.9, the jet version). The proof is fairly long and ...
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Milnor, Lectures on h-cobordism theorem: Lemma 5.9

In lectures on the h-cobordism theorem, Milnor writes of an isotopy $h_t:\mathbb{R}^n\to\mathbb{R}^n$, with $\mathbb{R}^n=\mathbb{R}^a\oplus\mathbb{R}^b$ in his notation, the following lemma: The ...
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The transversal pre-image of a manifold with boundary

I am reading "transversal pre-image of a manifold with boundary" from Differential Topology by M. Hirsch. I have some confusion regarding Theorem 4.2. on the page 31. Notice that the first ...
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Exercise of transversality on Hirsch's Differential topology

This is the second exercise of the chapter III (Transversality), where the principal theorems are about transversality of maps and jets, and an example of how this can be used to prove density of ...
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Coordinate representation of the local transversal

Let $S\subset M$ be compact embedded k dimensional submanifold,the embedding $\phi:B(0) \subset \Bbb{R}^{n-k}\to M$ be local transversal of $S$ at $p$(That is $\phi$ transversal to $S$ and $\phi^{-1}(...
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Loop in $S^2\times S^1$ with an orientable compact surface removed

Let $S$ be a closed orientable surface embedded in $S^2\times S^1$ assume that the projection of the surface $S$ on $S^1$ is not a point. I am trying to prove that exist a smooth loop $\gamma: S^1\to ...
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Existence of a local transversal section of a vector field

I am trying to prove the existence of a local transversal section, i.e. Let $U\subset\mathbb R^n$ be an open, $X:U\to \mathbb R^n$ a vector field of class $C^k$ and $p\in U$ a regular point of $X$, ...
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intersecting *semi-riemannian* manifolds

I read this interesting post: Intersection of manifolds. In geometry/topology one can consider intersection of manifolds (which need not be a manifold). I'm curious if one can do this in Semi-...
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Morse function induced on fibered product

Let $A,B$ and $C$ be three smooth manifolds. Suppose that $F:A\to C$ and $G:B\to C$ are smooth and transverse functions, making the fibered product $$S=A\underset{F,C,G}{\times}B=\{(x,y)\in A\times B\,...
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Is there any basis for ‪Transversal‬ matrix space?

Given That An N-By-N Matrix A, Is A Set Of Numbers With One From Each Row And Each Column Is Called A Transversal Matrix Given that an n-by-n matrix A, is a set of numbers with one from each row and ...
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Transversality of stable and unstable manifolds along a homoclinic orbit

Consider the system $$ \begin{cases} \dot{x}=f(x,\alpha), \\ \dot{\alpha}=0, \end{cases} $$ with $(x,\alpha)^T\in\mathbb{R}^{n+1}$. The system has a homoclinic orbit at $\alpha=0$ and we assume ...
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Intersection of lines with submanifold

I come across an interesting problem below which confuses me a lot. Let $L$ be a submanifold of $\mathbb{R}^n$ with codimension > 1 . Prove that: (1) if $x$ $\notin L$, for almost every line $l$ ...
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$L$ is submanifold of $\mathbb{R}^n$ with $\dim L\leq n-2$. [duplicate]

Suppose $L$ is submanifold of $\mathbb{R}^n$ with $\dim L\leq n-2$. Prove that If $x\notin L$, then for almost every line $l$ passing $x$, $l \cap L=\phi$; If $x\in L$, then for almost every line $l$ ...
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Transversality just on a single point but not in any neighborhood

I just learned the concept of Transversality. Briefly, let $f:M \to N$ be a smooth map and $A \subset N$ be a submanifold. If $K \subset M$ we write $f \pitchfork_K A$ to mean that $f$ is transverse ...
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John Lee's proof of Transversality Theorem

In John Lee's proof, I cannot understand why the equality holds in (6.9). (and I think only subset relation is needed in the proof.)
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How to show that the set of hyperplanes that are transverse to some projective variety is Zariski-open?

Let $V$ be a analytic/algebraic variety in $\mathbb{CP}^n$, that is $V$ is the vanishing locus of some homogeneous polynomial. Let $d = \dim V$. Everyone knows$^\mathrm{TM}$ that a "generic $k$-...
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question about global nonzero section of bundle

Suppose $M$ a manifold with dimension $m$ and $E$ a vector bundle whose based manifold is $M$. Assume $\mathrm{dim} E$ is $2m+1$, by Transversality theorem, seems I can obtain a global section without ...
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Dropping closed condition in Transversality Theorem

Most theorems stating that transversality is an open condition look like: We have manifolds $M$, $N$, and a submanifold $A\subseteq N$. Then assuming $A$ is closed in $N$, in the appropriate topology ...
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transversely intersecting spacetimes

A maximally symmetric causal diamond is a solution to Einstein's equation with a cosmological constant. Consider a causal diamond in a maximally symmetric spacetime for a ball-shaped spacelike region $...
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Understanding $T_{p}\mathbb{RP}^{2}$ ,tangent planes of his opens sets and transverse

I have to find two submanifolds $N_{1}$ and $N_{2}$ of dimension 1 of $\mathbb{RP}^{2}$ the real projective space such that $N_{1}\cap N_{2}\neq \emptyset$ and $N_{1}\pitchfork N_{2}$ . For this I ...
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Transversality if and only if a regular value

Let $f:M^{m}\rightarrow N^{n}$ a differential application and $S^{k}$ a submanifold of $N$. We say that $f$ is transversal to $S$ in $p\in f^{-1}(S)$ if $df_{p}(T_{p}M)+T_{f(p)}S= T_{f(p)}N$. If $S^{0}...
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Transversality in $2$-manifold

Let $X, Y$ be $2$-dimensional smooth manifolds without boundary, and $P$ be a disconnected $1$-dimensional boundaryless embedded submanifold of $Y$. Let $f\colon X\to Y$ be a smooth map that is ...
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How is The Number of Self-Intersections of a Compact Connected Riemann Surface Embedded in a Complex Surface defined?

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 3, Page 82. Let $M$ be a Complex Manifold of complex dimension two (i.e., $M$ is a Complex Surface). Let $S$ be a Compact ...
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Minimal sets, Perfect sets, Exceptional sets and Foliations

The first image is taken from Geometric Theory of Foliations, Book by A. Lins Neto and César Camacho, Chapter 3, Page 53. The second image is taken from Geometry, Dynamics And Topology Of Foliations: ...
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why does $ f| int(D)$ have only finitely many intersections with the knot?

We define a compressing disk of the knot $K$ (a smooth emmbedding from $S^1$ to $S^3$) to be a smooth map $f: D → S^3$ such that $f|∂D = K$ and such that $f| int(D)$ is transverse to $K$. Then $f| int(...
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The set of pseudo-gradient vector fields with the Smale property and a given set of trajectories is open (Proposition 3.4.3 in Audin-Damian)

In Audin Damian's proof of the Invariance of Morse Homology from vector field and Morse function, p.71, there's a proposition, where the Smale property is that the stable and unstable manifolds of the ...
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Transversality Condition and the Proof of Smale Theorem (Audin-Damian), Lemma 2.2.8 (Part 2)

In Audin and Damian, p.44-45, there seems to be a claim that one can prove transversality without directly showing the tangent spaces span the tangent space of the ambient manifold. In particular, in ...
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Stability of transversality

I'm having some troubles to prove that transverality is a stable property. First, some definitions. $F:M\rightarrow{N}$ and $A\subset N$. $F$ is transveral to $A$ $\ $ ($F\pitchfork A$) $\ $ if ...
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Holonomy of a Foliation that is Transverse to the Fibres of a Fibre Bundle

Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ be a Fibre Bundle and $\mathcal{F}$ be a $C^r$ (where $r \ge 1$) Foliation on $\mathbb{E}$ that is Transverse to the fibres of the fibre bundle $(\...
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Show that there exists a line that intersects with all lines $l$.

Consider a line $k$ in the projective space $P(\mathbb{R}^4)=\mathbb{R}P^3$ and 3 points $A_0,A_1,A_2\in k$. Let $B$ be an arbitrary point in $P(\mathbb{R}^4)$ but not on $k$ and $l$ be a "...
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$I_2(X,Z) = 0$ for $X,Z$ manifolds in $\mathbb S^k$ of complementary dimension

Suppose $f : X → \mathbb{S}^k$ is smooth, where $X$ is compact and $0 < \text{dim } X < k$. Then for all closed $Z \subset \mathbb{S}^k$ of dimension complementary to $X$, $I_2(X,Z) = 0$. The ...
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If $\alpha$ is an arbitrary simple curve that cuts transversaly a compact surface the set of common points it can be infinity?

MY ATTEMPT: Let $S^2=\{(x, y, z)\in \mathbb{R}^3| x^2+y^2+z^2=1\}$ be a compact surface. Defining for every $n\in \mathbb{N}$ a regular parametrized curve $\alpha_n:\mathbb{R}\rightarrow\mathbb{R}^3$ ...
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Which property of transversality enables Whitney's (weak) embedding? (Guillemin/Pollack)

In Guillemin & Pollack Differential Topology it says on P.49 with regards to Whitney's weak embedding theorem (i.e. every smooth $k$-manifold can be embedded in $2k + 1$ euclidean space): Why $N =...
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Paths transversal to countable collection of submanifolds

I am interested in the following claim, which is a bit counter-intuitive to me. I am wondering whether it is correct. Claim. Let $Y$ be an open (path-)connected subset of $\mathbb{R}^d$, and let $\{...
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3 votes
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Transversality in $\mathbb{R}^2$

Define, for some $a\in\mathbb{R}, f_{a}:\mathbb{R}\to\mathbb{R}^{2}$ by $f_{a}(p)=(p,a)$ and consider $N\subset\mathbb{R}^{2}, \ N=\{(x,x^2); x\in\mathbb{R}\}.$ Analysis of the transversality of $f_{a}...
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How can I find the value of x given multiple Parallel Lines and multiple Transversals?

Ive been given the following problem: I was unable to figure out how to solve the problem, and I was told the answer was the following; However, the program has failed to explain to me exactly how it ...
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Transverse submanifolds

Take $M$ to be a manifold and $N_{1},N_{2} \subset M$ to be two submanifolds such that $dimN_{1}+dimN_{2}=dimM$. Then $N_{1}$ and $N_{2}$ are transverse if and only if their intersection is infinite ...
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Genericity of an induced projection map

Let $X,Y$ be smooth manifolds, $S'$ a submanifold of $Y$, and $f:\mathbb{R}\times X\to Y$ a smooth function. Generically, we have that $f$ is transverse to $S'$, which implies that $S:=f^{-1}(S')$ is ...
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Almost all affine $k$-planes in $\mathbb{R}^n$ are transverse to a submanifold

I am trying to do the following differential topology exercise, I think I have come up with a solution but wanted to make sure it is correct Let $A$ be a smooth submanifold of $\mathbb{R}^n$. Then ...
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Intersections of sufficiently small sphere with fibers of a submersion are trasnverse

Suppose $(\mathbb R^n,0)\overset{f}{\to}(\mathbb R^k,0) $ is a real analytic map with $0\in \mathbb R^n$ an isolated critical point. I have read in many places the following assertion. For ...
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Transversal and intersection of two foliations

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two foliations of a manifold. We say that $\mathcal{F}_1\pitchfork \mathcal{F}_2$ if $T_p L^{(1)}+T_pL^{(2)}=T_p M$ for any $p\in M$, where $L^{(1)}$ and $L^{...
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Proving semi-Riemannian submanifolds are transversal

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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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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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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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 $N-S$ is dense in $N$, where $M,N$ are $m$ and $n$ dimensional $C^{\infty}$-manifolds, respectively. In this post the answering poster ...
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