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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Why do we require differential manifolds to be Hausdorff? [duplicate]

Among the requirements for a differential manifold $M$ is that it be connected and Hausdorff. What fails if a manifold is not Hausdorff?
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$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
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Lemma 1 p.205 (Differential topology - Guillemin and Pollack)

Lemma 1 : Let $S_1, ..., S_N$ be the covering of the closed intervalle $[a,b]$ in $\mathbb{R^1}$. Then there exists another cover $S'_1, ..., S'_M$ such that each $S'_j$ is contained in some $S_i$, ...
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Show that the natural copy of $\mathbb{R^{n-1}}$ inside $\mathbb{R^{n}}$ - namely, $\{(x_1, x_2,…, x_{n-1},0)\}$ - has measure zero

Question P.202 (Differential Topology - Guillemin, Pollack) : Show that the natural copy of $\mathbb{R^{n-1}}$ inside $\mathbb{R^{n}}$ - namely, $\{(x_1, x_2,..., x_{n-1},0)\}$ - has measure zero. ...
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Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
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Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis ...
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Huisken's distance comparison principle and type II singularities.

I've been reading Huisken's paper on his distance comparison principle and he remarked that in particular his theorem rules out the formation of type II singularities. These are singularities where in ...
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Curvature shortening flow for immersions

I was wondering if for an immersed curve in the plane, is it true that if the singularity points are evolved appropriately, then the curve becomes more embedded. And if so, would it eventually become ...
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Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
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Curvature shortening flow of embedded curves

QUESTION: I'm not sure how they proved part c in particular. Note that theorem 2.1 refers to Huiskan's distance comparison principle for evolving curves. I don't see why a separating boundary curve ...
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A manifold such that its boundary is a deformation retract of the manifold itself.

If we have a compact orientable manifold $M$, we know that $\partial M$ is not a deformation retract of $M$. This follows from Poincaré Duality or Stokes Theorem. If we take away compactness, this is ...
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$f$ integrable vs $\int_Af$ exist - in Spivak's Calculus on Manifolds

I am confused by the difference between "$f$ integrable" and "$\int_Af$ exist", in Spivak's notion of extended integral. Here's his definition. Note that it has a flaw: each $\varphi $, in the ...
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1answer
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Why should the source and the target map of a Lie groupoid be submersions?

In the definition of a Lie groupoid, the source and the target maps are required to be submersions. I want to know the reason for that. I write down definitions below. See also ...
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Is there a differentiable function on a closed subset of $\mathbb{R}^n$ that cannot be continued differentiably on an open superset?

Let $A \subseteq \mathbb{R}^n$ be closed with no isolated points and $f:A \to \mathbb{R}^m$. Suppose that for every point $x_0 \in A$ we have (at least one) matrix $L_{x_0}$ such that $$ \lim_{x,y \to ...
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Prove that General Linear Group is a topological subgroup.

First of all for $\mathbb{R}$ in my book it is written that: "$GL(n,\mathbb{R})$ is an open subset of euclidian $n^2$-space and that is the topology is given. Matrix multiplication is given by ...
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Is $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$ a differentiable submanifold?

Let $M=\{(x,y)\in \Bbb{R}^2 : x^2 = y^5\}$. Determine whether or not $M$ is a differentiable submanifold. I honestly couldn't get anything out of it. What is the standard approach to this ...
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$SL(n)$ is a differentiable manifold

Prove that $SL(n)=\{A\in \Bbb{R}^{n\times n}:\det(A)=1\}$ is a differentiable submanifold. The determinant function is smooth since it's a polynomial, and we have $\det^{-1}(1)=SL(n)$. So it ...
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Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a ...
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Let $U$ be an open set of $R^n$, and let $f: U \to R^n$ be a smooth map. If $A \subset U$ is a measure zero, then $f(A)$ is of measure zero.

Let $U$ be an open set of $R^n$, and let $f: U \to R^n$ be a smooth map. If $A \subset U$ is a measure zero, then $f(A)$ is of measure zero. Proof (Differential Topology - Guillemin and Pollack): $ ...
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Precisely what is meant by “$\pi_1(M)$ is torsion”?

I am reading a paper where one of the conditions for a Theorem to hold is "the group $\pi_1(M)$ is torsion", where here $M$ is a compact differentiable manifold. What is meant by the first homotopy ...
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Show that $f^*\omega = \det(df) \, dx_1\wedge\cdots\wedge dx_n$

Let $f:\Bbb{R}^n\to \Bbb{R}^n$ be a differentiable map given my $f(x_1,\ldots,x_n)=(y_1,\ldots,y_n)$, and let $$\omega=dy_1\wedge\cdots\wedge dy_n.$$ Show that $$f^*\omega = \det(df) \, ...
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Given an embedded $C^k$ submanifold of $R^n$, is the distance function $C^k$?

I am asking because I am wondering if studying smooth manifolds is the same as studying zero loci of smooth functions with non-vanishing gradient. Let me be a little formal for clarity: Let $M$ be a ...
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Criteria for boundary convexity of hypersurfaces in Euclidean space

I have a question about the relationship between two different formulations of the notion of boundary convexity, in the sense of Riemannian geomety. Let $M$ be an $n$-dimensional manifold with ...
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Tubular neighborhood by restricting the Riemannian exponential map

Let $M$ be a Riemannian manifold (possibly non-compact, possibly non-complete) and $N\subseteq M$ a smooth submanifold (possibly non-compact). Does there exist a continuous $\mu\colon M\rightarrow ...
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44 views

Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$

I think I managed to show this statements but I am not sure about it. Since this is common problem in differentiable manifolds I was wondering if anybody has (or may write) a solution. Let $X$ be a ...
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Understanding tensors

Locally in a chart, a tensor field looks like $$T= T^{i_1,...i_n}_{j_1,...,j_m} dx^{j_1} \otimes...\otimes dx^{j_m} \otimes \partial_{i_1} \otimes ... \otimes \partial_{i_n},$$ where ...
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Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
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Covariant and partial derivative commute?

I know that we have for a function $\Gamma: (-\varepsilon,\varepsilon)^2 \rightarrow M$ we have (at least I think I know that this is true) $$\nabla_{\frac{\partial \Gamma}{\partial s}} ...
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Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
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Show that $f:X\to \mathbb{RP}^n$ factors through $S^n\to\mathbb{RP}^n$ iff $f^*:H^1(\mathbb{RP}^n,Z_2)\to H^1(X,Z_2)$ is zero

Problem Show that (1) $f:X\to \mathbb{RP}^n$ factors through $S^n\to\mathbb{RP}^n$ iff $f^*:H^1(\mathbb{RP}^n,Z_2)\to H^1(X,Z_2)$ is zero (2) $f:X\to \mathbb{CP}^n$ factors through ...
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Homotopy type of intersection of complement of hyperplanes in projective space.

Let $U_i = \{x=(x_0 :… :x_n) \in \mathbb{P}^n(\mathbb{C}); x_i \neq 0 \}$ be the usual trivialization of the complex projective space. I have been trying to compute the homotopy type of all the ...
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Trigonometric parametrization of a genus g surface?

It is possible to find functions $\phi, \psi \in \mathbb{R}[sin(x), sin(y), cos(x), cos(y)]$, so that $S^2 = \phi( [0,1]^2)$ and $\psi( [0,1]^2)$ is a torus. Is it possible find, for any genus g, ...
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Understanding tangent space basis

Consider our manifold to be $\mathbb{R}^n$ with the Euclidean metric. In several texts that I've been reading, $\{\partial/\partial x_i\}$ evaluated at $p\in U \subset \mathbb{R}^n$ is given as the ...
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Continuity in definition of Induced Functional Structure

I have a really simple question, however I am confused. Bredon's Topology and Geometry gives definition of Induced Functional Structure as follows: Suppose $F_x$ is a functional structure on space ...
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119 views

Differential topology versus differential geometry

I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds: Analysis on manifolds, containing: definition of manifold, tangent space (as ...
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Is there a unique preferred connection on a general manifold?

I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined? Afais for Riemannian manifolds, there exists always exactly one Levi-Civita ...
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Thom space of unit circle

Say we embed $S^1$ into $\mathbb{R}^2$ as the unit circle. What is the Thom space $Th(i)$ associated to this embedding $i:S^1 \to \mathbb{R}^2$? By definition, the Thom space is the one point ...
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Symbol $\Gamma$ when talking about vector fields.

I noticed several times online that people tend to use the symbol $\Gamma(M,TM)$ when talking about the space of smooth vector fields on smooth manifolds. I find this totally confusing, as in ...
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32 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
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How small is Diff(M) compared to Homeo(M)?

Let $M$ be a smooth manifold. Is it always true that the group of diffeomorphisms is strictly contained in the group of homeomorphisms? (I know this is true for $\mathbb{R}^n$, but that is only a ...
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Problem of Arnold's book and covering spaces

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
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Open condition given by inequality on functions

Let's say we have two functions $f,g\in C^\infty(D)$, $D$ an open domain in $\mathbb{R}^2$. The condition $f(x,y)<g(x,y)$ is an open condition on $D$? With this I mean: do the points $(x,y)\in D$ ...
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Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
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58 views

Show Lie bracket left invariant

I want to prove from the definition that the Lie bracket $[X,Y]$ of two left-invariant vector fields $X,Y: G \rightarrow TG$ where $G$ is a Lie group is again left-invariant. Left-invariance ...
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Prove that $x+g$ is homeomorphism

Problem: Assume we have $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ of $C^1$ class with derivative bounded uniformly by some constant $M<1$. Consider ...
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Embedding of a smooth manifold

Let $M$ be a smooth, n-dimensional manifold. Prove that for every $k \leq n$ there exists an embedding $ \mathbb{R}^k \to M$. I'm having trouble visualising this. How can $\mathbb{R}^2$ be embedded ...
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Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
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Partial derivatives of all orders of linear map exist

If F is a linear map from R^n to R^m is it true that F is C^infinity, i.e. partial derivatives of all orders exist? My thought is that the answer should be "yes," because the derivative of F is just F ...
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Derivative group action [duplicate]

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. ...