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.

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Derivative group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
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Homotopy Equivalence and Local Coefficient Systems

Suppose I am computing the (co)homology of a nice space M using a local coefficient system G, i.e. $H_{*}(M, G)$. If M is homotopy equivalent to N, then M and N should have isomorphic (co)homology. ...
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Lagrange multipliers and critical points (differential form description).

On $M \times V^*$, where $M$ is a differentiable manifold (not necessarily equipped with a metric) and $V^*$ is dual to a vector space $V$, one can define a Lagrange function $F = f +v^*h(x)$ using ...
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Kinds of isometries preserving the curvature tensor

We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature. First I am ...
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Vector field from group action

Let $\Phi: G \times \mathbb{R}^4 \rightarrow \mathbb{R}^4$ be a group action where $G = \mathbb{R}/(2 \pi \mathbb{Z}).$ Then $$\Phi(\theta,(x_1,x_2,p_1,p_2)) = ( R(\theta) (x_1,x_2)^T, R(\theta) ...
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Mathematics is not a spectator's sport? [on hold]

The title is a sentence by John M. Lee, from his book "Introduction to Topological Manifolds". Indeed, I was wondering if one can learn mathematics in a passive way, just reading the books and ...
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26 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
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Show that $M$ is a differentiable submanifold

Problem. Let $f_i:\Bbb{R}^4\to \Bbb{R}, \,\, i=1,2,3,$ be defined by $$f_1(x_1,x_2,x_3,x_4) = x_1x_3-x_2^2\\f_2(x_1,x_2,x_3,x_4)=x_2x_4-x_3^2\\f_3(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3.$$ Then $M=\{x\in ...
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Proof of Whitney intersection theorem

I know "Lectures on the h-cobordism theorem" by Milnor lists a proof. Unfortunately, the proof is subtle, intricate, and lengthy, that is, not succinct and elegant enough (for Milnor, of course.) Is ...
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Show smoothness of this map

Let $S^3$ be the sphere identified with the subset $\mathbb{C}^2$ as $\{(x,y) \in \mathbb{C}^2; |x|^2+|y|^2=1\}.$ Then I want to show that the map $\phi: S^3 \rightarrow \hat{\mathbb{C}}$ is ...
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Group action and smooth manifolds

I was wondering if it is for a compact (i.e. Hausdorff) smooth manifold $M$ sufficient to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
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63 views

Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
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Boileau-Orevkov on the intersection of a complex curve with 3-sphere

Motivation: Feel free to skip. Let $\Sigma$ be a complex curve in $\mathbb{C}^2$. If $\Sigma$ is transverse to a 3-sphere $S^3 \subset \mathbb{C}^2$ of radius $r$, then $\Sigma \cap S^3$ is a link, ...
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Jeffrey Lee 2.11 Show there is a nice map $s:TTM \to TTM$ satisfying several properties

I'm not sure this problem makes any sense on several levels, but here is the question verbatim: Find natural coordinates for the double tangent bundle $TTM$. Show that there is a nice map $s:TTM ...
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Parametrizing the time an element stays in an open subset

Let $X$ be a topological space (If it helps anything, we can assume $X\subseteq\mathbb{R}^n$ or $X$ being a smooth manifold.) and $U\subseteq [0,1]\times X$ an open subset. Does there exist a ...
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decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g-1)$ pants bounded by 3 geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
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Vector fields spanning the tangent bundle at each point

Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each ...
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43 views

page 4 of Milnor's book on Morse Theory

I have a stupid and probably naive question about one line in the book of Milnor about Morse theory. What does exactly means if $v \in T_pM$ then there is an associated vector field $\tilde v $ ? I ...
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1 parameter subgroups and Lie groups

I was just reading some lectures notes (that are not online available unfortunatley) on Lie groups and found that sometimes the author just says if he wants to prove something for all Lie group ...
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54 views

Orientability and Hypersurfaces

I got stucked in this problem: Show that: i) Every embedded closed hypersurface $S$ is orientable. ii) Every differentiable hypersurface defined by a regular cartesian equation $\ g(x_1,..., x_n)=0$ ...
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Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
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50 views

Smooth function, which separates between a closed and a open set.

Let $M$ be a smooth manifold, $O\subseteq M$ an open subset and $B\subseteq M$ a closed subset, such that $closure(O)\subseteq interior(B)$ I think there must exist a smooth function $f\colon ...
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55 views

Possible mistake in *Curves and Singularities*, 2nd ed., by Bruce & Giblin, p. 74

Page 74 in Bruce & Giblin's Curves and Singularities, 2nd ed. contains the following passage: Let $f: I \to \mathbb R$ be smooth and define $\phi: I \times \mathbb R^2 \to \mathbb R$ by $\phi ...
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86 views

Theorema egregium violated in dimension $n \ge 4$?

Gauß showed that for surfaces in $\mathbb{R}^3$ the Gaussian curvature ( = sectional curvature) is invariant under local isometries. This is known as the thema egregium. Now in another question ...
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63 views

Meaning of “locally homeomorphic to $\mathbb{R}^{n}$”

I am fairly new to differential geometry and approaching it with a physics background (in the study of general relativity), as a result I'm having a few struggles with terminology etc, so please bear ...
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Integral of generator $H^{2k}(\mathbb CP^k)$ over $\mathbb CP^k$ is 1

I'm trying to give a proof of the Hirzebruch signature theorem from a differentiable viewpoint (i.e. using de Rham cohomology). The proof works, except for one small step. We know that ...
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33 views

(Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture 1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere. 2.) ...
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resonance in pde

(I migrated the question from overflow b/c I think it belongs here instead.) About a year back I was in Professor Henry Mckean's office and he told me alot of interesting things. One thing he told me ...
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Mistake in the definitions of the linking number.

I am looking into the definition of the linking number. I've considered these two definitions. Consider a link $L$ with components $K_1$ and $K_2$, and respectively their embeddings $\gamma_1$ and ...
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Vanishing of the first Chern class of a complex vector bundle

Suppose that $E\to M$ is a $\mathbb{C}^n$-bundle with a metric. This is equivalent to saying that there exists a chart $\{U_\alpha\}$ of $M$ and $\phi_{\alpha,\beta} \colon U_\alpha\cap U_\beta\to ...
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1answer
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Tilting sine function to get countably infinite nonregular values?

Let $f: \mathbb R \to \mathbb R$. A nonregular value $y$ of $f$ is any value such that not all $x \in f^{-1}(y)$ are regular. A point is regular if the Jacobian at it is surjective, in this case, has ...
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Why is this map $H^1$?

I have the following proposition (taken from Klingenberg's Lectures on Closed Geodesics): Let $\pi: E \rightarrow S$ and $\mathcal{O} \subset E$ be a finite dimensional fibre bundle over the ...
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Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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Lagrange's Equation on a Manifold

I know that, if $L: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}$, then the Euler-Lagrange equation is: $$ \nabla_x L - (\nabla_{\dot{x}}L)' \equiv 0$$ In trying to ...
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The fundamental vector fields of a principal bundle are vertical.

Let $p:P\to M$ be a principal $G$-bundle. To each $A$ in the Lie algebra of $G$ corresponds a fundamental vector field $A^*$ on $M$ defined by $$A^*_u=\frac{d}{dt}|_{t=0} u(exp(tA))$$ How can we see ...
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What can be said about the space of vector fields for which a given, say $C^1$, function is a Lyapunov function?

I am learning Morse homology and I have been thinking about the following observation. One way of doing, say finite-dimensional, Morse theory is by fixing a Morse function $f\in C^{\infty}(M)$, where ...
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Proving smoothness of left-invariant metric on a Lie Group

Assume $G$ is a Lie group. The standard construction of a left invariant metric on $G$ goes as follows: Take an arbitrary inner product $\langle,\rangle_e$ on $T_eG$ and define $\langle u , ...
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On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
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Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
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About separation property of hypersurface

Let N be a complete Riemannian manifold and M be a complete hypersurface in N. M is said to have separation property if N\M is disjoint union of 2 connected open sets in N. Under what reasonable ...
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Existence of diffeomorphism through convergence in Hausdorff distance

I'm reading a book and have come across something that I cannot verify or fix. The assumption is that $\Omega_1, \Omega_2, ...$ is a sequence of connected open sets in $\mathbb{R}^n$ that converge in ...
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Tangent and normal spaces of submanifold of fixed-rank matrices

Let $m \geq 2$. The subset $X$ of $m \times 2$ matrices with rank $1$ is a (smooth) submanifold of $\mathbb{R}^{m\times 2}$. Let $A$ be in $X$. I know from a more general statement that the tangent ...
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Is it true that all $k$-submanifolds of a $m$-manifold are open subsets of some closed $k$-submanifold?

Let $M$ be a $m$-dimensional (smooth) manifold. I know that $m$-submanifolds of $M$ are exactly the open subsets of $M$. Is it true that all $k$-submanifolds of $M$ are open subsets of some closed ...
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A “parallel manifold” is always orientable

I want to solve the following problem from Spivak's Calculus on Manifolds: Let $M$ be an $(n-1)$ dimensional manifold in $\mathbb{R}^n$. Let $M(\varepsilon)$ be the set of end points of normal ...
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Locally ringed space locally isomorphic to a *closed* subset of $\mathbb{R}^n$

To me it's more natural to think of e.g. a tetrahedron as a closed subset of $\mathbb{R}^3$ than as a "manifold with corners" in the traditional sense -- i.e., locally isomorphic to open subsets of ...
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Incorrect statement in a proof of the transversality theorem?

I'm reading through Morris Hirsch's book on differential topology, and he makes the following offhand statement. Suppose k is a compact subset of a manifold U, and V is a vector subspace of R^n. If a ...
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Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
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Local Submersion Theorem - Differential Topology of Guillemin and Pollack

Local Submersion Theorem : Suppose that $f:X \to Y$ is a submersion at $x$, and $y=f(x)$. Then there exist local coordinates around $x$ and $y$ such that $f(x_1,...,x_k)=(x_1,...,x_l)$. That is, $f$ ...
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Introductions to Ehresmann connections and Chern-Simons forms

I am looking for introductory texts on Ehresmann connections and Chern-Simons forms. I seek detailed, hands-on presentation. Please, recommend sources that employ a differential forms approach rather ...
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Relation beween transition functions of a principal fiber bundle and its dual

What is the relation between transition functions of a principal fiber bundle and its dual? As an example, consider the transition map of the frame bundle on a manifold of dimension $n$ as the ...