Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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3
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211 views

How to show the existence and uniqueness of the pullback connection in vector bundles?

There is the following result: If $D$ is a connection on a vector bundle $E$ over $N$ and $φ$ is a smooth map from $M$ to $N$, then there is a pullback connection on $φ^*E$ over M, determined ...
6
votes
2answers
409 views

Why the tangent bundle is Hausdorff?

I was reading the lemma 4.1 in "J.M.Lee - Introduction to smooth manifolds" which says that given a smooth $n$-manifold $M$, then the tangent bundle $TM$ is a smooth $2n$-manifold. If $\pi: ...
0
votes
0answers
21 views

Homotopic curves

I'am studying differential forms and I've arrived at the theorem of invariancy of integrals of closed forms over homotopically equivalent curves. I came up with a problem: If I take a curve with ...
0
votes
1answer
35 views

why is $\dfrac{dr}{r~d\theta} = \cot \psi$?

why is $\dfrac{dr}{r~d\theta} = \cot \psi$ ? Extracted from Ordinary Differential Equations, Garrett Birkhoff, in the chapter of Linear Fractional Equations (First order Differential Equations). ...
0
votes
3answers
236 views

unit speed curves and frenet serret

Let us assume that $\alpha(s)$ is a unit speed curve with $\kappa > 0$. I'm trying to find the vector function $w(s)$ such that $$T' = w \times T,\quad N' = w \times N,\quad B' = w \times B.$$ I ...
0
votes
0answers
23 views

Prove: A unit speed curve $\alpha(s)$ with $\kappa>0$ is rectifying if and only if $\frac{\tau}{\kappa}=as+b$

Given a curve $\alpha$, the plane determined by {$T,B$} is called the rectifying plane. A unit speed curve is said to be rectifying if for all $s$ and for some constant $p$, $\alpha(s)-p$ lies in the ...
1
vote
2answers
30 views

Rank of Jacobian Matrix for the Stereographic Projection

With the definition $S^{n} = \{\ \mathbf{x} \in \mathbb{R}^{n+1}\ | \ ||\mathbf{x}|| = 1\ \}$, and the function $\ f:\mathbb{R}^{n} \to S^{n} \setminus \{ (0,...,0,1) \}$ defined by: $f(\mathbf{u}) ...
1
vote
2answers
44 views

What are some interesting uses for/motivations of projective spaces?

I have trouble motivating myself to think about real projective spaces, for instance. Are there any cool results about them? Are there any motivating examples?
3
votes
2answers
31 views

Is any smooth deformation of a metric in dimension 1 conformal?

Consider $(S^1, g)$ where $S^1$ is the unit circle and g is a metric. Now consider the metric $$ \tilde g := f g $$ where f is a smooth positive function. Since in 1 dimension this is the only smooth ...
1
vote
0answers
14 views

Regarding the axis of screw motion for a space curve.

The axis of the accompanying screw motion of a curve $c(s)$ at any point $c(s_0)$ is the line in the direction of the Darboux vector $\tau(s_0) T(s_0) + \kappa(s_0)B(s_0),$ through the point $$P(s_0) ...
1
vote
1answer
36 views

Is the associated bundle construction a bifunctor?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and ...
2
votes
1answer
40 views

Complex Hopf Fibration

The Hopf construction gives a circle bundle $p$ : $S^{3}$ → $\mathbb{CP}^1$. The equation of a 3-sphere in $\mathbb{R}^4$ is $X^2+Y^2+V^2+W^2=R^2$, where $R$ is the radius of the 3-sphere. We may ...
6
votes
3answers
181 views

For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?

How do I see that for $n$ even, the antipodal map of $S^n$ is homotopic to the reflection$$r(x_1, \dots, x_{n+1}) = (-x_1, x_2, \dots, x_{n+1}),$$and therefore has degree $-1$? Thanks for your time. ...
8
votes
2answers
67 views

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 ...
3
votes
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 $q(x) = \mathbb{R}x =$ ...
8
votes
2answers
302 views

On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where ...
2
votes
0answers
49 views

Cohomology algebra generated by Steifel-Whitney classes and dual classes subject to defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
2
votes
0answers
23 views

Hermitian Structure on $\mathcal{O}(1)$ line bundle over $\mathbb{P}^{n}$

I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can ...
1
vote
0answers
30 views

Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that ...
1
vote
0answers
26 views

Why is the backwards curve needed for the inverse of a parallel transport?

Dearh math.stackeschange-community, I'm at a loss with the following problem: Let $I=(a,b)$, then the inverse of the parallel transport $P_{s,t}^\gamma$ from $s \in I$ to $t \in I$ along the curve ...
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votes
0answers
18 views

Orientation bundle of a manifold [closed]

How can i show that orientation bundle of any smooth manifold ( whether or not orientable manifold) is orientable.
2
votes
1answer
14 views

Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $ \gamma (t) = (sin(t)+2,0,t) $ on XZ plane for $ t \in R $, ...
1
vote
0answers
28 views

derivative of a projection matrix

The projection onto a parametrised vector $v(\lambda)$ is $P_v = \frac{vv^{T}}{v^{T}v}.$ Its complement is $$P = I-\frac{vv^T}{v^{T}v}.$$ I've got an expression containing this complementary ...
1
vote
0answers
16 views

Showing that $(\mathbb{R}, \mathscr{F})$ and $(\mathbb{R},\mathscr{F_1})$ are diffeomorphic but $\mathscr{F}\neq \mathscr{F_1}$

Background $M$ is locally Euclidean with dimension $d$ if $M$ is hausdorff and every point in $M$ has a neighborhood homeomorphic to $\mathbb{R{^d}}$. If $U\subset M$ is open and connected and $\phi$ ...
3
votes
0answers
39 views

Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
2
votes
0answers
28 views

A step in proof of Bishop-Gromov Theorem

I am reading the proof of Bishop-Gromov's comparison theorem in Schoen and Yau's Differential Geometry book. They do it using the Jacobi fields approach. There is a step which I have trouble ...
5
votes
1answer
106 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
0
votes
0answers
18 views

Number of Frenet Frames

How can I show that a regular curve in $\mathbb{R}^n$ has $2^n$ Frenet frames? I guess intuitively I'm thinking that since a Frenet frame is a moving reference of $n$ orthonormal vectors and each ...
1
vote
1answer
21 views

Are the symplectic leaves of a Poisson manifold submanifolds?

In "Introduction to Mechanics and Symmetry" by Marsden and Raţiu it is written, in chapter 10, page 347, example b, that "[s]ymplectic leaves need not be submanifolds". In "Lectures on Poisson ...
4
votes
0answers
28 views

Not understanding the foliation structure of a Poisson manifold

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, ...
0
votes
1answer
24 views

Christoffel symbols of $S^n$ in polar coordinates

Consider the usual local polar coordinates $\theta_1, \theta_2,..., \theta_n$ on $S^n$. We were taught about Christoffel symbols today and I am trying to see what the Christoffel symbols of $S^n$ ...
3
votes
2answers
46 views

How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
1
vote
0answers
39 views

Killing vector field for Witten complex?

I am reading a classical paper by Atiyah Bott "The moment map and equivariant cohomology". In paragraph "Relation with Witten complex" (at the very beginning of this paragraph) they claims that ...
1
vote
0answers
22 views

Hessian of the Stereographic projection

Consider the stereographic projection from the sphere $S^n$ onto $\mathbb{R}^n$, and take the usual local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from its embedding in ...
3
votes
1answer
28 views

Another way of describing a maximal torus

Consider the Lie group $SU(2)$. A maximal torus for $SU(2)$ is $$T=\left\{\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}:\theta\in{\Bbb R}\right\},$$ and its Lie algebra is ...
2
votes
1answer
20 views

Proving subset of regular surface - hyperboloid - is a regular surface

I have stumbled upon this in differential geometry dealing with regular surfaces: We define the following surface (a hyperboloid) as $ K = \{ (x,y,z) \in R^3 | x^2+y^2-z^2 = 1 \} $ and ...
0
votes
2answers
32 views

Can a surface of revolution be built from a self-intersected curve?

I'm reading "Differential Geometry of Curves And Surfaces" of Manfredo Do Carmo. There's a point in his book about Surfaces of Revolution which confuses me a lot. Here is the part: The part ...
2
votes
0answers
34 views

Level sets and lagrangian submanifolds

I would like to find all regular values for a map on a symplectic manifold such that the level sets are lagarangians. Precisely, the following example : The $4$-dimensional symplectic manifold ...
1
vote
1answer
22 views

Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?

When I read Lee's Riemannian Manifolds : An Introduction to Curvature, I am confused by the red line in the picture below. Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?
0
votes
1answer
33 views

Rewriting line integral for complex-valued function

Context: Suppose $f = \phi + i\psi$ is continuous and $\gamma(t):[a, b] \to \mathbb{C}$ is a curve. Then we define the integral of $f$ along $\gamma$ to be $$ \int_\gamma\!f = \int_a^b ...
2
votes
1answer
58 views

Holomorphic Frobenius Theorem

I'm trying to understand a proof of the Holomorphic Frobenius Theorem using the smooth version as seen in Voisin's Complex Geometry book: (pg 51) ...
4
votes
0answers
55 views

Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
7
votes
4answers
987 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
4
votes
1answer
37 views

Equivalence between constant and positive metric and usual $\Re^3$ metric

I'm trying to answer the following question: Is any positive and constant metric in $\Re³$ equivalent to the usal metric defined as $$ds² = dx² + dy² + dz² \tag{*}\label{1} $$ with $ds = ...
1
vote
2answers
74 views

Quadratic form equals zero

We have a quadratic form $x^TAx$ where $x$ is a vector in $\mathbb R^n$ and $A$ is an $n \times n$ real symmetric matrix. Define M to be the set: $$M=\{x \in \mathbb R^n| x^TAx=0\}$$ and a ...
0
votes
1answer
30 views

Understanding the first fundamental form of a surface, how the parametrization doesn't matter.

The following is an excerpt from Pressley's Elementary Differential Geometry on the definition of the first fundamental form. However, there are some parts of this concept that I'm unclear about. It ...
1
vote
2answers
23 views

Finding the error in the surface area of a cube. when length = 3, error= ${1\over 4} $

Find the approximate error in the surface area of a cube having an edge of length 3ft if an error of ${1 \over 4}$ in. is made in measuring an edge I have to do this by using differentials and ...
0
votes
1answer
74 views

The tangent space to a particular subset of $\Bbb R^n$ at a particular point

Let A be a real symmetric $n \times n$ matrix and define the set M by: $$M = \{x \in \mathbb R^n \, | \, Ax \cdot x = 0\} .$$ What is the tangent space to $M$ at the origin? I think it should ...
1
vote
1answer
30 views

Differential geometry of projective bundles

Can someone give me a reference about projective bundles from a differential-geometric point of view? I am not very familiar with algebraic geometry. I would like, for example, some theory about when ...
0
votes
1answer
28 views

Angle between surface normal and arc normal of Frenet Serret frame

Is the the angle $\varphi$ indicating deviation of geodesy ( in Meusnier's thm) always $ \pi/2 $ for an asymptotic line with $ k_n =0 $ ? (I could not find such a statement in textbooks I referred ...