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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Proving that the general linear group is a differentiable manifold

We know that the the general linear group is defined as the set $\{A\in M_n(R): \det A \neq 0\}$. I have a homework on how to prove that it is a smooth manifold. So far my only idea is that we can ...
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Reference request: Vector bundles and line bundles etc.

I am interested in learning algebraic geometry and I talked to one of my professors today (who is, in part, an algebraic geometer) and he recommended I understand the analytic analogue of the ideas in ...
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1answer
311 views

Extension of Riemannian Metric to Higher Forms

I've been reading about Riemannian manifolds, and have come across a comment that says that for a metric $g$ on an $N$-dimensional manifold $M$, considered as a bilinear map $$ g:\Omega^1(M) \times ...
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Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
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Gradient in differential geometry

I am a graduate student in physics trying to learn differential geometry on my own, out of a book written by Fecko. He defines the gradient of a function as: $ \nabla f = \sharp_g df = g^{-1}(df, ...
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61 views

Is there a nowhere $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
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218 views

Differential Geometry: Is a closed disk a surface?

An open disk is clearly a surface, in the sense that it is locally homeomorphic to a part of $\mathbb{R}^2$. But what about a closed disk, even though it still looks like a surface, I am starting to ...
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284 views

Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant.

Let $r(s)$ be a curve parametrized by arc length, and $\kappa,\kappa',\tau$ are non-zero. Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant. The teacher ...
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1answer
136 views

Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $ \Bbb R^n \to \Bbb R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of ...
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Equivalent condition for non-orientability of a manifold

I've just came across this question, which gives us a great tool for showing that smooth manifold is non-orientable. Namely Thm. If $M$ is a smooth manifold and there are two charts ...
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1answer
96 views

Can one build a homology theory using submanifolds and their boundaries?

Consider a manifold $M$, and denote by $\Delta _p M$ the set of all submanifolds of dimension $p$ (with or without boundary) of $M$. Define $G_pM$ to be the free abelian group generated by $\Delta_p ...
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tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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Use implicit function theorem to show $O(n)$ is a manifold

In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal ...
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1answer
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Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
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1answer
170 views

The Affine Property of Connections on Vector Bundles

Given any two connections $\nabla_1, \nabla_2: \Omega^0 (V) \to \Omega^1 (V)$ on a vector bundle $V \to M$, their difference $\nabla_1 - \nabla_2$ is a $C^\infty (M)$-linear map $\Omega^0 (V) \to ...
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How I could define a inner product in the characters in $SL(2, \mathbb R)$

I have homework in a course on Lie groups, in which I must show that $(\pi_m,\mathbb C_m[x,y])$ are the only irreducible representations of finite dimension in SL(2,ℝ). Here $ C_m[x,y])$ is the vector ...
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2answers
105 views

What is the space curve with curvature and torsion obeying

$ \kappa = \cos s, \tau = \sin s $ and passing through (1,0,0), TNB triad identity matrix? previous link When numerically computed it looks like a catenoid surface of revolution for all ...
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Immersion of $\mathbb{R}$ to $\mathbb{R}^2$

I have no idea how to prove that the set $\{(x, |x|): x\in\mathbb{R}\}$ is not the image of an immersion of $\mathbb{R}$ into $\mathbb{R}^2$ For example If $f(t)=(t^3, |t|^3)$ then $f'(0)=(0, 0)$.
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1answer
979 views

Proof of the second Bianchi identity

I'm asked to prove the second Bianchi identity: $$\nabla_{[e}R_{ab]c}^{\;\;\;\;d}=0$$ using the fact that: $$(\nabla_a \nabla_b -\nabla_b \nabla_a)\omega_c=R_{abc}^{\;\;\;d}\omega_d$$ For every diff. ...
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1answer
182 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
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906 views

Generating orthogonal axes on a spline

I use a Catmull-Rom spline, each section built from a set of cubic functions p = x(t), y(t), z(t). I was looking at Frenet Formulas but it's not working as ...
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79 views

number of points of tangency of the zero divergence vector field and the equator of the sphere.

Let $V$ be vector field on the sphere $S^2$ and $\operatorname{div} V=0$. What is the minimum number tangency points of this vector field and the equator of the sphere?
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What does it mean to say a boundary is $C^k$?

I need a explanation on what does it mean to say a boundary is $C^k$. Can anyone help me please. And also need some explanation on how to straighten boundary ?
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Non surjectivity of the exponential map to GL(2,R)

I was asked to show that the exponential map $\exp: \mathfrak{g} \mapsto G$ is not surjective by proving that the matrix $\left(\matrix{-1 & 0 \\ 0 & -2}\right)\in \text{GL}(2,\mathbb{R})$ ...
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famous space curves in geometry history?

For an university assignment I have to visualize some curves in 3 dimensional space. Until now I've implemented Bézier, helix and conical spiral. Could you give me some advice about some famous ...
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501 views

Surface where number of coordinate charts in atlas has to be infinite

In the definition of a parametrised surface $S$, for every point in the surface, $p \in W \subseteq S$, where $W$ is open, there exists a coordinate chart or patch , $F :U\to \mathbb{R}^n$ that maps ...
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745 views

Total Derivative and Multilinear Functions

So I'm starting to work through Spivak's Calculus on Manifolds and I'm having a little trouble verifying some of the claims made in the book problems. To review: Given a function ...
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2answers
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Jacobian matrix rank and dimension of the image

I am having problems with the following question: What is the relation between the rank of the Jacobian matrix of $f$ (which is continuously differentiable) and the dimension of the image of $f$? ...
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1answer
712 views

A local diffeomorphism of Euclidean space that is not a diffeomorphism

Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in each point) that is not a ...
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201 views

What does it mean for the group of rotation matrices to have a “manifold structure”

I have just been exposed to rotation matrices, and it is showing extremely strong connection with group theory and differential geometry which I am both totally inept at. Can someone in simple term ...
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Is $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ diffeomorphic to $S^2$?

I was working on Problem 5-1 of Smooth Manifolds by Professor John Lee, and it lead me to wanting to show that $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ is diffeomorphic to $S^2$, and that is ...
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1answer
106 views

Volume of neighborhood of the curve

Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|<r\}$. How can i show ...
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Does positive definite Hessian imply the Jacobian is injective?

Suppose $f(x):\mathbb{R}^n \mapsto \mathbb{R}$ is an infinitely differentiable function. If $\nabla^2 f(x)$, the hessian of $f$ is positive definite everywhere, does this imply that the gradient(first ...
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1answer
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How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
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Proving $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$

Let $X,Y$ be vector fields. $L_X$ is the Lie derivative and $i_X$ is the contraction of a $k$-form. I am really stuck on how you could prove the identity $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$. Update: I ...
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Confirming an error in a textbook: O'Neill's Semi-Riemannian Geometry

I'm working on the following exercise in O'Neill's "Semi-Riemannian Geometry": (Page 53, 12) Let $b$ be a symmetric bilinear form on $V$. The nullspace of $b$ is $N = \{v : b(v,w) = 0, \, \forall w ...
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Why do all solutions to this equation have the same form?

In this paper on page 45, the authors state that Let's assume we know that $$ w \times dw = d\varphi + \sum_{j=1}^3\alpha_jdx_j, \tag{1}$$ where $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, ...
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Nilpotent infinitesimals comparison

I'd like to understand better the advantages and disadvantages of various approaches to nilpotent infinitesimal numbers and their application to differential geometry in the context of physics and ...
4
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2answers
974 views

Reconstructing space curves from its curvature and torsion

I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
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1answer
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Why can we think of the second fundamental form as a Hessian matrix?

Let $f: U \rightarrow \mathbb{R}^3$ be an immersion that parametrizes a piece of a surface, and let $(h_{ij})$ be the matrix for the second fundamental form of that surface. According to pg. 70 of ...
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Zeroes of exact differential forms on compact manifold

Let $M$ be a $n$ dimensional compact differentiable manifold. I would like to show that any exact differential form of degree $n$ vanishes at at least one point. I think it is a generalization of the ...
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Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
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240 views

Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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1answer
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What is the definition of $dx$

I have just started to study differential forms. I don't yet fully understand the definition of what a differential form is (it's a $p$-times covariant tensor field) but I know that if $U$ is an open ...
3
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2answers
159 views

Surface with non-zero mean curvature means orientable

Let $M$ be a surface in $\Bbb R^3$ with non-zero mean curvature for every point. How could I show that this implies that $M$ is orientable? By our definition, orientable means that an unitary, normal ...
3
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2answers
452 views

Parametric curve on cylinder surface

Let $r(t)=(x(t),y(t),z(t)),t\geq0$ be a parametric curve with $r(0)$ lies on cylinder surface $x^2+2y^2=C$. Let the tangent vector of $r$ is $r'(t)=\left( 2y(t)(z(t)-1), -x(t)(z(t)-1), ...
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1answer
755 views

The square root of positive definite matrix

Let $M$ be the manifold of real positive definite $n \times n$ matrices, define a mapping $i:A \to \sqrt A$ (where $A\in M$ and $\sqrt A$ means the unique positive definite square root of $A$). Please ...
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1answer
404 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
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1answer
84 views

Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$

Prove: $h \in T_xM \iff \operatorname{dist}(x+\epsilon h,M) = o(\epsilon)$ For $M$ a smooth manifold in $\mathbb{R}^n$, $h\in\mathbb{R}^n$, $x \in M$. I know that $T_xM = \operatorname{Im}(Df) = ...
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3answers
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Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 - \{(0,0)\})$ but then realised that for better ...