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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$\mathbb{R}^1$-bundle $\xi$ possesses Euclidean metric iff $\xi$ represents an element of order $\le2$

The set of isomorphism classes of $1$-dimensional vector bundles over $B$ forms an abelian group with respect to the tensor product operation. How do I see that a given $\mathbb{R}^1$-bundle $\xi$ ...
4
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1answer
53 views

Any $\mathbb{R}$-linear mapping $X: C^\infty(M, \mathbb{R}) \to \mathbb{R}$ with $X(fg) = X(f)g(x) + f(x)X(g)$ given by $X(f) = Df_x(v)$?

Let $M$ be a smooth manifold, and let $C^\infty(M, \mathbb{R})$ denote the collection of smooth real valued functions on $M$. For $x \in M$, how do I see that any $\mathbb{R}$-linear mapping $X: ...
5
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2answers
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Smooth manifold $M$ is completely determined by the ring $F$.

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
7
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2answers
63 views

If $M$ is compact, every maximal ideal in $F$ arises in this way as a point of $M$?

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
3
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1answer
42 views

Collection of smooth real valued functions on smooth manifold has ring structure.

For any smooth manifold $M$, how do I see that the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and that every point $x \in M$ determines a ...
5
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1answer
86 views

Tensor product for vector bundles is commutative, associative, and has an identity element?

How do I see that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element?
7
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67 views

How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of tangent $2$-planes? [duplicate]

A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a subbundle of dimension $k$. How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of ...
3
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2answers
46 views

Subset $V$ of projective space is open iff $q^{-1}(V)$ is open?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinate space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$by $q(x) = \mathbb{R}x =$ ...
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0answers
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Extending Morse-Smale pair from submanifolds?

The following proposition is extracted from Audin & Damian's Morse Theory and Floer Homology, Proposition 4.6.3: Let $(f,X)$ be a Morse-Smale pair on $V$ (a submanifold of $W$). Then there ...
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Diffeomorphism between $\mathbb{P}^n$ and the submanifold of $\mathbb{R}^{(n+1)^2}$ consisting of certain matrices?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinates space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$ by $q(x) = \mathbb{R}x ...
4
votes
1answer
44 views

Equivalence between smooth and topological fiber bundles

All manifolds in this post are hausdorff and second-countable. Is it true that two smooth fiber bundles with same fiber, base manifold and structure group (that is a Lie group $G$ of diffeomorphisms ...
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1answer
23 views

A trivial vector bundle with a riemannian metric has an isomorphism with the trivial bundle that is an isometry on each fiber.

The problem: Demonstrate that If $\xi^n=(E,\Pi,M)$ is a trival vector bundle with a riemannian metric, then there exist an isomorphism $\psi:E\to$ M x $R^n$ Such that $\psi $ is an isometry on each ...
4
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1answer
31 views

If $M$ is Riemannian, then $\kappa_f \oplus f^*TN \cong TM$, where $\kappa_f$ is built out of kernels of the $Df_x$?

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. I know how to construct a vector bundle $\kappa_f$ built out of the ...
6
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2answers
246 views

Unique factorization of manifolds?

I wonder if there is a result on the unique factorization of manifolds. Call a topological manifold to be indecomposable if it is not homeomorphic to a product of manifolds of positive dimension. Is ...
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1answer
38 views

Constructing a vector bundle built out of kernels of the Jacobian?

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. How do I construct a vector bundle $\kappa_f$ built out of the kernels of ...
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2answers
29 views

Can derivative of a smooth norm be zero?

Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map. Is it true that its differential at (every non-zero point) ...
14
votes
1answer
124 views

Exists homeomorphism which carries each fiber isomorphically to itself, composition?

Let $\mu$ and $\mu'$ be two different Euclidean metrics on the same vector bundle $\xi$. How do I see that there exists a homeomorphism $f: E(\xi) \to E(\xi)$ which carries each fiber isomorphically ...
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2answers
54 views

Smoothness of transition maps of fiber bundle

A smooth fiber bundle with fiber a smooth manifold without boundary $F$ and structure group $G$ a Lie group of diffeomorphisms of $F$ is a smooth surjective map $p:E\to M$ between manifolds, with a ...
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336 views

Is the tangent bundle of $S^2 \times S^1$ trivial or not?

As the question title suggests, is the tangent bundle of $S^2 \times S^1$ trivial or not? Progress: I suspect yes. If I could construct three independent vector fields, I would be done. But I'm not ...
0
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0answers
44 views

Constant rank theorem for manifolds with boundary

In this post, " $M^n$ is a manifold" stands for " $M$ is a $n$-dimension smooth manifold with or without boundary". Definition. A neat map between smooth manifolds is a smooth map $f:M^m \to N^n$ ...
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1answer
33 views

why does the sequence have a convergent subsequence

Back again! I only have two more questions, I promise... help is much appreciate as I seem to have found myself stuck and pretty much turned in a blank worksheet to my professor. He says these types ...
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45 views

commutivity of suspension and quotient space

Let $(A,A_0)$ be a $CW$-pair such that $A_0$ is a $CW$-subcomplex of $A$. Let $\Sigma$ be the suspension. Is $\Sigma(A/A_0)$ homotopy equivalent to $\Sigma A/\Sigma A_0$ or not?
7
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1answer
50 views

Boy's surface, visualization of the preimage of self-intersection locus as graph on projective plane

For the immersion of the projective plane in $\mathbb{R}^3$ with one triple point, what does the preimage of the self-interaction locus as a graph on a projective plane look like?
3
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1answer
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Construction of an immersion of $T^3$ − point in $\mathbb{R}^3$?

Let $p\in T^3$. How do I construct an immersion of $T^3\setminus\{p\}$ in $\mathbb{R}^3$?
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27 views

Hopf link and degree of a map

I'm considering a problem of computing the degree of a map $\varphi: S^{1} \times S^{1} \rightarrow S^{2}$ defined as $$\varphi(x, y) = \frac{\gamma_{1}(x)-\gamma_{2}(y)}{|\gamma_{1}-\gamma_{2}|}$$ ...
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8 views

property of a difference measure on a convex set

Let $D$ be a difference measure on a convex symmetric set $S\subseteq{}\mathbb{R}^{n}$ with a centre point $m$ (for example an equilateral triangle or a circle in $\mathbb{R}^{2}$). $D$ does not need ...
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1answer
33 views

Direct Limit of Grassmannians

Let $X$ be a topological space and $G_n(\mathbb{C}^m)$ be the space of vector subspaces of $\mathbb{C}^m$ of codimension $n$. Let $G_n(\mathbb{C}^\infty):=\bigcup_{m=n}^{\infty}G_n(\mathbb{C}^m)$ ...
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Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold

I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem: Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...
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0answers
21 views

Is infinity always a regular value of rational function from $S^1\to S^1$

$f$ is a rational function $S^1\to S^1$, where $S^1$ is the one point compactification of $\mathbb R$. Is $\infty$ always a regular value of $f$?
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vote
1answer
45 views

Find grid points interior to a closed curve

Consider a simply structured grid $G$ represent a domain $\Omega$, e.g. Cartesian (2D/3D) or Polar\Spherical etc. For simplicity the following discussion is about 2D case. Let $\Gamma\subset \Omega$ ...
7
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1answer
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Tangent manifold $D(M_1 \times M_2)$ is canonically diffeomorphic to the product $DM_1 \times DM_2$.

Let $M \subset \mathbb{R}^A$ and $M_2 \subset \mathbb{R}^B$ be smooth manifolds. How do I see that the tangent manifold $D(M_1 \times M_2)$ is canonically diffeomorphic to the product $DM_1 \times ...
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2answers
33 views

What is the difference between n-dim ($n<\infty$) Banach space and $R^n$?

What is the difference between n-dim ($n<\infty$) Banach space and $R^n$? I feel they are same ,no matter under homeomorphism isomorphism or diffeomorphism. In fact, I feel they are not ...
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1answer
35 views

Clarification about differentiable manifolds

When checking the transition maps for differentiability in order to determine if a manifold is differentiable, do we fill in any removable singularities (i.e. simplify the function composition before ...
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1answer
43 views

What's the cohomology of disjoint union of two circles

I am computing the cohomology of $T^2$ by Meyer-Vietoris sequence. $T^2$ can be seen as the union of two open sets U and V s.t. U and V are diffeomorphic to a cylinder respectively. Thus U$\cap$V is a ...
5
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1answer
181 views

Non-diffeomorphic structures on the sphere

How many smooth structures are there on $S^2$, $S^3$, and $S^4$ up to diffeomorphism? I looked around and couldn't find an answer; two books I have say different things on the subject. I know one of ...
0
votes
1answer
12 views

How large is the subgroup of diffeomorphisms which preserves a vector field?

Let $M$ be a smooth manifold. Let $X \in \Gamma(TM)$ be a vector field on $M$, which vanishes at a finite number of points. (Every smooth manifold admits such a vector field). Consider the subgroup ...
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1answer
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A product of smooth manifolds together with one smooth manifold with boundary is a smooth manifold with boundary

Suppose $M_1, \dots M_k$ are smooth manifolds and $N$ is a smooth manifold with boundary. Then how do I see that $M_1 \times \dots \times M_k \times N$ is a smooth manifold with boundary, ...
2
votes
1answer
25 views

Clarifying a point in Guillemin and Pollack on a certain submersion

In Guillemin and Pollack, given $f:X \rightarrow \mathbb{R}^{M}$ we are given a function $F:X \times S \rightarrow \mathbb{R}^{M}$ (where $X$ is a manifold and $S$ is an open ball in $\mathbb{R}^{M}$) ...
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1answer
41 views

Given a smooth map, find another similar map with a differential that is never zero (Guillemin and Pollack 2.3.8).

I have spent hours on this problem without getting far. The actual statement is as follows: Suppose that $f:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a smooth map, $n>1$, and let $K \subset ...
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1answer
40 views

Differentiating function with norm

Let E be a vectorial space of finite dimension. $E^*=E\setminus \{0\}$. $$f:E^*\rightarrow E^*$$ $$x\rightarrow \frac{x}{\langle x,x\rangle}$$ Is this function $C^1$? How should I proceed? ...
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1answer
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Lefschetz number of a transformation of the sphere.

In differential topology the Lefschetz number of an automorphism of a compact manifold is the oriented intersection number of the graph of that automorphism with the diagonal. I would like a proof or ...
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1answer
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Proving an embedding

Note: below the word embedding is supposed to also be an immersion as in its differential topology definition. Question: $$ f : \mathbb{R}^2 \rightarrow \mathbb{R}, f(x,y) = x^3 + xy + y^3 + 1 $$ ...
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0answers
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Show that Inflection points in $f^{-1}(0)$ are those in the which has $f_{yy}f_x^2 - 2f_{xy}f_x+f_{xx}f_y^2 =0.$

Let $f:\mathbb{R}^2\to\mathbb{R}$ a function of class $\mathcal{C}^2$ and $(x_0,y_0)\in\mathbb{R}^2$ so that $Df(x_0,y_0)\not=0$. Then exists $g:I\to J$, with $I$ and $J$ open intervals that contain ...
2
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1answer
35 views

Derivative vanishes $\implies$ Locally Constant

This is a little lenghty, and there's probably an shorter way to see this, but I want to prove the following: Let $f:N^n\to M^m$ be a smooth map from between smooth manifold and $\{U,x\}$, $\{V,y\}$ ...
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1answer
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Proof that the range of a map is determined by its behaviour on the boundary.

Let f be a mapping from an open neighbourhood of the 3-dimensional unit ball to the 2-dimensional plane. Suppose that f is smooth (infinitely continuously differentiable on its domain) and regular ...
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3answers
35 views

Differential of a matrix function

Let $n \geq 1$. Calculate the differential of the function G: $$G: M_n( \mathbb{R}) \times M_n( \mathbb{R}) \rightarrow M_n( \mathbb{R})$$ $$(A,B)\rightarrow A^tBA$$ My first thought was to add ...
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0answers
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Is it possible to construct explicitely a homologue between two smooth simplicial cycles $A,B$ in a manifold that satisfy $\int_A w = \int_B w$?

Suppose that $M$ is a compact manifold, and $[A]$ and $[B]$ are two cycles in the (real coefficients) homology of $M$ that are represented by smooth simplices $A$ and $B$. If we assume that $\int_A w ...
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1answer
28 views

Topological Manifold is Manifold with Empty Boundary

I want to show that every $n$ topological manifold $M$ is an $n$ Manifold with boundary where $\partial M=\emptyset$. i.e. every chart $(U,\phi)$ maps to an open set $V\subseteq\mathbb{H}^{n\circ}$ ...
2
votes
2answers
33 views

why topological conjugacy does not preserve ergodicity?

I want to know why topological conjugacy does not transfer ergodicity from one dynamical system to another?and if we change the maps from continous to C1-differentiable will the problem solve or not?
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Some questions about linear operator in Brouwer degree.

Let $f:\Omega\rightarrow R^n$ is $C^1$, $\Omega$ is bounded open subset of $R^n$. And $f(x_0)=\theta$, $J_f(x_0)\ne 0$.$J_f$ is Jacobi of $f$, $\theta=(0,...,0)$. Then ,how to show : $f'(x_0)$ has ...