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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Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
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Almost Every Hyperplane is Transverse to $M$

Let $M$ be an $n$-dimensional manifold embedded in $\mathbb{R}^{n+1}$. I am trying to show that almost every hyperplane in $\mathbb{R}^{n+1}$ is transverse to $M$. To show that I would like to prove ...
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Derivative of the inclusion of a submanifold

I know there are other questions similar to this one, but I just want you to tell me if what I'm doing is rigth and how to improve it. The problem is the following: (I'm using the definitions by ...
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How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
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Why can't we usually speak of partial derivatives if the domain is not open?

In the book by Guillemin & Pollack they define a function $f$ from an open subset $U\subset R^n$ to $R^m$ to be smooth if it has continuous partial derivatives of all orders. Then they say ...
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Parametrizing Walks on Sphere and Torus

This question is very underdeveloped, but I was wondering if there was a map from the sphere to the torus which preserves length of closed curves? I was just thinking about taking a walk on a ...
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Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
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Generalizations of the product neighborhood theorem

As far as I know, the Product Neighborhood Theorem for smooth manifolds states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization ...
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55 views

Proving a Certain Smooth Map $S^n\rightarrow S^n$ is a Diffeommorphism

I am given a smooth map $f:S^n\rightarrow S^n$, for $n\geq 2$, whose differential is injective at each point. I am asked to prove that it is a diffeomorphism. Since the differential is injective ...
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Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
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57 views

Show That $\dim H_m(\partial M;\mathbb{R})$ is Even

A student asked me this. Suppose that $M$ is a compact, orientable $n$-manifold with boundary. It is a fact that for each $k$ with $0\leq k\leq n$ the vector spaces $H_k(M;\mathbb{R})$ and ...
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a question related with morse theory [duplicate]

Show that there exists no smooth function $f:\mathbb{R}^2→\mathbb{R}$,such that $f(x,y)\geq 0$ for any $(x,y)\in\mathbb{R}^2$, with exactly two critical points$(x_1,y_1)\in\mathbb{R}^2$, ...
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Why is every derivation a vector?

We can see the vectors of the tangent space $T_pM$ to a smooth manifold as velocities of curves. This is elaborated here. Each velocity $\gamma'(0)$ corresponds to a derivation $D_{\gamma}(f) = (f ...
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2answers
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Is the $G$-action on a principal $G$-bundle proper?

Let $G$ be a Lie group. If $G$ acts properly and freely on a manifold $P$, then it is well-known that $P \to P/G$ form a principal $G$-bundle. I would like to know the converse: namely Question: if ...
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A question about the existence of a smooth function [duplicate]

Does there exists a smooth function $f: R^2 \rightarrow R$, such that $f(x,y)\ge0$, for any $(x,y) \in R^2$, and $f$ has exactly two critical points $(x_1,y_1), (x_2, y_2) \in R^2$ with ...
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Help understanding a proof in differential geometry

I was reading John Milnor's Topology from the Differentiable Viewpoint and there's a proof of the fundamental theorem of algebra at the end of the first chapter that I don't fully understand. I can ...
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What is $T^*(\mathbb{A}^1)$?

Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
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$\mathbb{R}^2$ is a Retract of $\mathbb{R}^5$

Looking through some old qualifying exams, I noticed the question: Let $X\subseteq \mathbb{R}^5$ be homeomorphic to $\mathbb{R}^2$. Prove that $X$ is a retract of $\mathbb{R}^5$. I'm not even sure ...
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Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
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Avoiding vertical vectors in tangent spaces.

Let $i: [0,1]\hookrightarrow\mathbb{R}^3$ be a smooth embedding. Can I find arbitrary small perturbations $j$ of $i$ s.t. $j(0)=i(0)$, $j(1)=i(1)$ and $j'(t)$ is never a multiple of $(0,0,1)$? More ...
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Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
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Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
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Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
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Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?

Depending on interpretation, there may be an assumption missing from Exercise 6.1.4(a) in Liviu I. Nicolaescu's Invitation to Morse Theory: Suppose $f : \mathbb{R} → \mathbb{R}$ is a proper Morse ...
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50 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
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natural projection on a slice

I'm currently studying Warner's book "Foundations of Differentiable Manifolds and Lie Groups". Within the proof of the Frobenius Theorem he is constructing a slice $S$ of a coordinate system ...
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25 views

Extensions of diffeomorphisms of $S^2$ and the connectedness of $\text{Diff}^+(S^2)$

In this MO question by Daniel Moskovich, he claims that the fact that every diffeomorphism of $S^2$ extends to a diffeomorphism of $D^3$ implies that $\text{Diff}^+(S^2)$, the group of ...
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Accepted symbol (or way of writing) “A is a subset of B or B is a subset of A”

I am looking for a concise way to write the statement "$A$ is a subset of $B$ or $B$ is a subset of $A$". The context is the Grassmannian and two elements $A,B\in G_k(\mathbb R^n)$ in it. The two ...
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Connected sum in an ambient space

Let $M$ be a smooth c connected $m$-manifold, $N_1$, $N_2$ two smooth disjoint connected $n$-submanifolds with boundary, both contained in the interior of $M$ (if necessary). Assume that $M\setminus ...
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Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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Does there exist a dipole field on $S^2$ differing by at most a minus sign between antipodal points?

Consider the two-sphere $S^2 \subset \mathbb{R}^3$. By a dipole field on $S^2$, I mean a continuous function $f \colon S^2 \to S^2$ such that (1) $x$ is perpendicular to $f(x)$ for all $x \in S^2$ ...
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Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same ...
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Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
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Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...
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Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
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Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
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Intuition behind the definition of a derivative by Lang

In Serge Lang's Introduction to Differentiable Manifolds he says that a function $f:U\to F$ is differentiable at a point $x_0\in U$ if there exists a linear map $\lambda$ of $E$ into $F$ such that, if ...
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What is the pushforward of a function (not a vector)

If we have two manifolds $M$, $N$ with the map $f:M \to N$, then this induces a map between their tangent spaces $f_*:T_pM \to T_{f(p)} N$. By duality, another map exists $f^* : T^*_{f(p)}N \to ...
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Question on framed bordism classes definition

I was reading recently about cobordism, and in specific about the Thom-Pontraygin theorem which states $\pi_{k}(S^n)$ is isomorphic to the cobordism classes of framed $n$-manifolds in $R^k$. In ...
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Reference requestion: Existence/construction of bump functions

I'm not much of an analyst myself, but I've time and time again come across proofs which require knowledge of the existence of bump functions. However, I've never studied them, so I'm missing ...
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Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
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homotopying a function to itself by a regular homotopy

I wonder whether there is a smooth function $f:\mathbb{S}^{1}\rightarrow \mathbb{R}^{1}$ which may be homotoped to itself by a regular homotopy $H(x,t)$, i.e. by a smooth ...
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Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
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Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $ R^2-\{0 \} $ is zero, ...
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80 views

Change of Coordinate Formula for Differential Forms

Let $M$ be a manifold, $x$ local coordinates on an open set $U$, $y$ local coordinates on an open set $V$. In addition, let $(x, \alpha)$ and $(y, \beta)$ be two induced bases for the common part of ...
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Linking number of a pair of circles

Definition Given disjoint manifolds $M$, $N\subset \Bbb{R}^{k+1}$, the linking map $\lambda: M\times N\to S^k$ is defined by $\lambda(p, q) = (p - q)/||p - q||$. If $M$ and $N$ are compact, oriented, ...
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Turning a torus inside-out

Smale's paradox is now famous, and great videos can be found illustrating it. Similarly, there is a video showing how to turn a torus inside-out. The solution seems to be simpler, but is the proof ...
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Metric Tensors and its Taylor Expansion in Normal Coordinates

With metric tensors of the unit sphere in normal coordinates, their Taylor series for $p\in S$ near the north pole $N$ can be written as follows. $$g_{rr}(p) \equiv 1; g_{r\theta}(p) = g_{\theta ...
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Reference about Gauss-Bonnet-Chern theorem.

I would like to get some references which explains Gauss-Bonnet-Chern theorem and its original proof by Chern. I tried to read his paper published in 1944 "A Simple Intrinsic Proof of the Gauss-Bonnet ...
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Metric Tensors in Geodesic Normal Coordinates

Consider the unit sphere and its north pole $(0, 0, 1)$. My question is how to write the metric tensor $g_{ij}$ in geodesic normal coordinates. I know that metric tensors are defined as inner product ...