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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Natural isomorphism between linear space to bilinear space

Let $V$ and $W$ be (not necessarily finite-dimensional) vector spaces. Show that there is a natural isomorphism (meaning an isomorphism that can be described without reference to a basis) between ...
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42 views

Isometry: the dot product and derivative dot product.

In the image below what is meant by the items within the squares: Box 1. Derivative f is the function from tangent plane p to to tangent plane f(p)? Box 2. What is the meant by dot product ...
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Topology of the cuspoidal cubic

Let $C$ denote the set of solutions to $zy^2 = x^3$ inside of $\mathbb{C}P^2$. Someone told me that this space is homeomorphic to the pinched torus (or pinched sphere depending on how you pinch) - ...
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Applications of the Hurwitz Theorem on Number of Automorphisms?

So I am giving a talk in which we'll prove the semi-famous Hurwitz Theorem: Let $S$ be a compact Riemann surface of genus $g \geq 2$. Then $|Aut(S)| \leq 84(g-1)$, the group of holomorphic ...
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69 views

How to embed a square as a submanifold in the Euclidean plane?

One can give a square a differential structure, namely by transferring the smooth structure of a circle to it. But then a square in not a sub-manifold of $\mathbb{R}^2$. But the strong Whitney ...
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Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn't know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then ...
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54 views

An equivalence relation for atlas on an abstract set X

Let $X$ be an abstract set. A chart for a set $X$ is a bijection $\phi$ from a subset $U$ of $X$ to an open subset of $\mathbb{R}^n.$ Denote $(U, \phi).$ Two charts $(U, \phi)$ and $(U', \phi')$ will ...
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69 views

Einstein manifolds and topology

Given a Riemannian manifold $(M,g)$ with Ricci tensor $ R_{mn} = k g_{mn} $. Suppose the Ricci scalar you get is $$ R > 0 $$ What can you tell about the manifold $globally$ ? In particular, can ...
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26 views

Is the total space of a principal bundle parallelizable?

Given a smooth $G$-principal bundle $P \to M$, is $P$ in general parallelizable as a manifold? That is, is the tangent bundle $TP \to P$ trivial? In the case of Klein geometries, and more generally, ...
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29 views

Pullback on differentials of the 2-Sphere

I'm considering the diffeomorphism of the 2-Sphere given by the antipodal map and the pullback given by this map on the differentials $d\theta$ and $d\phi$. Let $\psi$ be such a map. My intuition ...
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22 views

Directional derivative near the bounder .

Let $\varphi:B^n\rightarrow B^n$ be a differential homeomorphism, $B^n$ is unit ball of $R^n$ , and $\overrightarrow{n}$ is outer normal vector of $B^n$. I feel that $$ \frac{\partial ...
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29 views

How to show that in a 6 dimensional manifold $\ast_6 A = - J \wedge A$ for $A^{1,1}$ primitive $1,1$ complex form and $J$ k\"ahler form

Given a 6 dimensional manifold, of complex dimension 3, take the Hodge star operator $\ast_6$ and a primitive (1,1)-form $A_2$ (i.e. such that $J \wedge \ast_6 A = J_{mn}A^{mn}=0$ and also $J\wedge J ...
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1answer
50 views

Is the normal bundle of a hypersurface in a orientable manifold trivial?

Suppose $N^{n+1}$ is a orientable smooth manifold, $M^n$ is a smoothly embedded hypersurface in of $N^{n+1}$. Is the normal bundle of $M^n$ trivial?
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58 views

An example of a compact topological manifold which has one cover?

I have been studying compact topological manifolds lately, in particular the $n$-sphere, $S^n$. The reason $S^2$ cannot be covered by one chart is because it is closed and bounded (and hence, by ...
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76 views

Prove that the unit circle centred at the origin is a submanifold of $R^2$.

I am having trouble applying the definition of submanifold. In this question, how do I select an atlas for $R^2$ and then how do I choose an atlas for $S^1$? This is the definition I am using:
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43 views

Least number of embedded hypersurfaces which separate a compact manifold

Suppose that $N^{n+1}$ is a connected closed smooth manifold. Define $$S(N^{n+1}) = \min \bigl\{ k \mid \forall (M^n_i), \, \bigcup_{i=1}^k M^n_i \text{ separates } N^{n+1} \bigr\}.$$ Here $M^{n}_i$ ...
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201 views

Homotopically trivial $2$-sphere on $3$-manifold

Let $S^2$ be an embedded sphere in a $3$-manifold $M^3$ such that $[S^2]$ is trivial in $\pi_2(M)$. Can we find an embedded disk $D^3$ in $M$ such that $\partial D^3=S^2$?
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Prove the tangent space at a point $x$ of the $n$-sphere is the space $\{v \in \mathbb{R}^{n+1} : v\cdot x=0\}$

I can see why this is true but I'm not sure how to prove it, any help would be appreciated. Prove that the tangent space $TS^{n}_{x}$ at a point $x$ on the $n$-sphere $S^{n}:=\{x \in ...
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110 views

Prove that $df_{x}: TM_{x} \rightarrow TN_{f(x)}$ is a well-defined map

I was wondering if someone could help me with the following problem, any help would be greatly appreciated. Let $f:M \rightarrow N$ be a $C^{\infty}$ map between smooth manifolds. Given $x \in M$, ...
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20 views

Prove that if (M, U) is a smooth n-dimensional manifold and p ∈ M, then there is a chart x: U → Rn such that x(p) = 0.

This is a textbook problem from the book Introduction to Manifolds by Tu(Pg-41). I have no idea how to go about this problem. Any hints/suggestions would be greatly appreciated.
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1answer
65 views

integrate over a cube given some differential form

What is process of integrating a differential form given some cube (hyperdimensional obejcts)? I read a lot qualitative problems on this, but seem to find rare examples on how to compute such ...
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Laplacian of a function at points of extrema

Consider a smooth function $f$ on a compact manifold $M$. Let $p$ be a point where $f$ is maximum and $q$ be a point where $f$ is minimum. Do we necessarily have $\Delta f|_p \leq 0$ and $\Delta f|_q ...
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How does the forgetful functor from $\textsf{Man}$ to $\textsf{TopMan}$ factor under the ternary factorization system?

Here, $\textsf{TopMan}$ refers to the category of topological manifolds (second-countable, Hausdorff, locally Euclidean topological spaces) with continuous maps for morphisms; and $\textsf{Man}$ ...
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59 views

De Rham cohomology group of the Klein Bottle

I need to compute all the cohomology rings of the Klein Bottle. I want to apply the Mayer-Vietoris sequence. Here I'm using the same good open cover suggested by the Wikipedia page :-) It's ...
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28 views

Is $T_{(r,0)}(TM)\cong T_rM$?

Where(r,0)$\in T_rM$, $M$ is an manifold.Is $T_{(r,0)}(TM)\cong T_rM$?
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34 views

de Rham Cohomology of the complex projective spaces

I want to compute the de Rham cohomology of the complex projective spaces $P^n_{\mathbb{C}}$. I know what the result is, and I've seen many posts in the forum asking the same thing. The problem is ...
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21 views

Prove that $x\in\text{Bd}(U)\iff f(x)\in\text{Bd}(V)$

$\newcommand{\Fr}{\text{Bd}}$ I denote $\text{Bd}(A)$ the boundary of the et $A$. Let $U,V\subset \mathbb R^n$ open and $f:\overline{U}\longrightarrow \overline{V}$ an homeomorphism. Therefore ...
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39 views

tensor product of sheaves is not a sheaf

I was looking for an example of two sheaves that satisfy the gluing and uniqueness axioms but whose tensor product is a presheaf, but not a sheaf (doesn't satisfy uniqueness, for example). Thank you ...
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126 views

Are PL embeddings homotopic to smooth ones?

In particular, I'm interested in the case we have a PL embedding $f : S^1 \to M$ (for a smooth manifold $M$): can it be homotoped to a smooth embedding? I'm not very familiar with PL stuff, so I may ...
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Prove that the map $F:\mathbb{R}\to S^1$, $F(t)=(\cos t,\sin t)$ is $C^\infty$.

Why isn't smoothness of $(\cos t,\sin t)$ as a map from $\mathbb{R}$ to $\mathbb{R}^2$ enough? IS it because when we have $S^1$, the codomain is changed? But that shouldn't affect smoothness, ...
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How did the smoothness of map into R^m transfer to the smoothness of map into R

The following is a proof from the book Introduction to manifolds by Tu ( Page 63) I did not understand the highlighted step. Please explain.
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Generalized Jordan-Brower separation theorem

Suppose $M^{n+1}$ is a closed connected smooth manifold and $N^n$ is a closed connected smooth embedded submanifold of $M^{n+1}$. What's the weakest condition under which the Jordan-Brower ...
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Fiber monodromy of complex multiplication

Consider the map $f: \mathbb{C}^2 \to \mathbb{C}$, with $f(z,w) = zw$. I read a remark that if we take a circle around the origin (say the usual loop starting and ending at 1) and we lift it to a map ...
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59 views

Faulty Argument: Chern number of U(1)-bundle over $T^2$ is zero?

Consider a $U(1)$-bundle $P$ over the two-dimensional torus $T^2$. Given a local curvature $F$, We can compute the first Chern number $c_1(P)$ by considering a rectangle $R_{\epsilon}$ in the center ...
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37 views

A sphere with a hair in R^3 is not locally euclidean at the root of the hair.

This is a problem from the book Introduction to manifolds by Tu (Pg-57 problem 5.2) The idea being referred to here is about connectedness i.e if we remove the root of the hair then it would ...
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1answer
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proving a particular subset of a Riemannian manifold is closed using continuity

I have two Riemannian manifolds, $(M,g)$ and $(\widetilde{M},\widetilde{g})$ and two maps $\varphi, \psi : M \to \widetilde{M}$, which are both local isometries. I am trying to show that the set $A = ...
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36 views

Smooth diffeomorphism and $C^1$ diffeomorphism

Given two smooth manifolds $M$、 $N$, and a $C^1$ diffeomorphism $f:M \rightarrow N$. Is there a smooth diffeomorphism between $M$ and $N$?
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Difference between $\mathbb{T}^n$ and $S^n$ and applications to dynamics (Euler angles and configuration manifolds)

I am struggling to see the difference between the $n$-sphere and the $n$-torus. We define $\mathbb{T}^n = S^1 \times S^1 \times \cdots \times S^1$, where the Cartesian product is taken $n$ times. I ...
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3answers
57 views

Construct a diffeomorphism that extends the identity map

Is there a diffeomorphism $f:[1,+ \infty) \rightarrow [1,3)$, such that $f$ restricted on $[1,2]$ is the identity map? I think its true, but I don't know how to construct one.
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51 views

Definition of Riemannian Metric

Let $(M, g)$ be a Riemannian manifold. Standard definitions of a Riemannian metric $g$ states that $g$ specifies a symmetric, bilinear, positive definite form on each tangent space $T_{p}M$ that ...
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Defining differentiability on a topological manifold

I am currently watching a series of lectures which are a part of International Winter School on Gravity and Light 2015 (by Prof. Frederick P. Schuller). In one of the lectures on differentiable ...
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idempotent bundle homomorphism has constant rank

I am reading Introduction to differential topology, T.H Bröcker & K. Janich and I could not solve the following exercise: Let $(E,\pi, X) $ be a vector bundle over a connected space $X$, let $f: ...
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Atiyah-Singer index theorem

I have seen the cohomological form of the index theorem usually stated in the following form: $$ \int_X \varphi^{-1}\left(\operatorname{ch}([\sigma(P)])\right).\operatorname{todd}(TX\otimes\mathbb C) ...
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64 views

Regular values and maps of degree 0

There is an elementary phenomenon in differential topology, that I've never quite understood: It is well known that the mapping degree (Brouwer degree) $\operatorname{deg}(f) = ...
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1answer
41 views

Definition of Normal Bundle and little exercise

I need to show that, given a manifold $M$ and its diagonal $\Delta\subset M\times M$, we have $T\Delta\cong\mathcal{N}_{\Delta|M\times M}$, where $T\Delta$ is the tangent bundle and ...
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47 views

Diffeomorphism on $S^2$ to $S^2$ with at least two fixed points

Let $f:S^2\to S^2$ diffeomorphism orientation-preserving such that for every fixed point of $f$, $1$ is not an eigenvalue of $df_p$. Then $f$ admits at least two fixed points. I'd to ...
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29 views

Relation between Thom class and closed Poincaré dual

Let $S,M$ be oriented manifolds without boundary such that $i:S\to M$ embedding and $i(S)$ is closed in $M$. $$ \eta_{S|M}:=\frac{TM_{|S}}{TS} $$ is the normal bundle of $S$ in $M$. I want to find a ...
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The Normal Bundle of a level set is trivial

Let $M,N$ be manifolds and $f:N\to M$ an embedding. For every $p\in N$, $df_p:T_pN\to T_pM$ shows that $T_pN\subset T_pM$. For every $p\in N$, define the normal space of $N$ in $M$ at $p$ as $$ ...
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Derivations having local character

Given a (smooth) manifold, it is known that derivations $D:C^\infty(U) \to C^\infty(U)$ on a chart $(U,\kappa)$ are equivalent to a vectorfield on $U$, i.e. to an element $X \in \Gamma(TU \to U)$. The ...
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Coadjoint orbit and orbit of adjoint representation

Let $Ad_g$ be the adjoint representation of a matrix group, so something like $\mathfrak{gl}.$ Then $Ad_g(\xi) = g \xi g^{-1}.$ Then I want to show that $Ad_g^*(x)= g^{-1}x g.$ As far as I know, the ...