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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A form of chain rule to differentiate the flow of a vector field on a manifold

I am reading the proof for a theorem about connections on a manifold, but I'm not comfortable with the fancy language of vector bundles and flows of vector fields I think. I wonder if there's an easy ...
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1answer
11 views

f differentiable map of finite dimensional vector spaces, with derivative injective. Why is f injective?

Suppose A and B are finite dimensional vector spaces, $U\subseteq{A}$ is an open subset, $a\in U$ and $f:U\rightarrow B$ is $C^\infty$ with $(Df)_a$ injective. I need helping showing that there exists ...
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17 views

Clarification about the definiton of a metric on a vector bundle

Let $E$ be a vector bundle over a manifold $M$. By definition, a metric on $E$ is a function $g:E \times _M E \to M \times \mathbb{R}$. In wikipedia, they say $g$ is a bundle map. However, it is not ...
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Lie group acting on itself defines a manifold

Let $G$ be a Lie group acting on itself as $\phi(h)(g)= L_h(g)$ as a left translation. Then we can consider the cotangent lift of this action, namely $\Phi: G \times T^*G \rightarrow T^*G$ as ...
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Gauss Curvature Equation

One can find the following form of Gauss Curvature Equation in most introductory books on manifold, for example Jeffrey Lee's Introduction to Differential Geometry p564: $$<R(V,W)X,Y> = ...
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Role of connection in a differentiable manifold

It's almost clear that a linear connection can be viewed as Vector bundle that makes a PARALLEL TRANSPORT .But I don't understand how they mean to connect vectors. What is the role of connection in ...
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32 views

What's the derivative of a map defined on manifolds?

I'm going through Warner's book on differentiable manifolds. On page 8 he defines what it means for a map $f: U \subset M \to \mathbb R$ to be differentiable: $f$ is differentiable iff $f \circ ...
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34 views

How do we address a function whose values are again functions?

One may call a function whose values are functions simply a function-valued function. But is there a canonical name for such an object? A $k$-form on some open $A \subset \mathbb{R}^{n}$ is an ...
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179 views

Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
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14 views

Relationship between Laplacian and Taylor expansion for 2nd derivative

I am working on converting a solution to a certain PDE from working on a regular 2D grid to work on a 3D triangular mesh. In the 2D scenario the 1st and 2nd derivatives are, of course, approximated ...
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27 views

Vector fields and tangent vector fields?

I am wondering if there are times when people would call a tangent vector field simply by a vector field? Are not these two concepts different? For example, a vector field assigns (say) to each ...
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31 views

The arc length in Riemannian geometry is well defined (independent from the choice of the coordinates)

Let be $(M,g)$ a connected Riemannian manifold and $p,q \in M$. If $ \phi : [a,b] \rightarrow M$ is $C^\infty$ we define the arc-length of the curve $\phi$ as the quantity: $$J(\phi )= \int_a^b ...
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39 views

green's second identity application

I need to use the green's second identity in order to prove the following equality: $$ \int_{\mathbb{R}^2} \ln (\sqrt{x^2+y^2})\Delta f = -2\pi f(0)$$ where $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ ...
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26 views

Proving the Leibniz Rule for Lie Derivatives of tensor fields.

I am learning some Differential Geometry on my own in preparation for a course I'm starting in October, and one of the exercises in the notes I'm using is to check that the Lie Derivative satisfies ...
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17 views

Is this end-point map surjective

Consider the differential equation: $\frac{d U_s}{dt} = (a + w(s)b)U_s$ where $w$ is some unknown, smooth, real and bounded function on the interval $[0,T]$ and $a,b \in \mathfrak{su}(n)$. Let ...
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1answer
20 views

Metric compatibility of dual connection

Let $(M,g)$ be a Riemannian manifold with Levi Civita connection $\nabla$. Then $\nabla$ satisfies a compatibility condition: $(\nabla_ZX,Y)+(X,\nabla_ZY)=Z((X,Y))$ where $(\cdot,-)$ is a Hermitian ...
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1answer
44 views

Determinant structure of symplectic matrix

I want to show that if $\lambda$ is a real eigenvalue of a symplectic matrix $A$ then its char poly is of the form $\det(A-\mu id) = (\lambda-\mu)(\frac{1}{\lambda}-\mu) \det(\hat{A}- \mu id) $ where ...
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25 views

Show that the normals to a parameterized curve all pass through the z-axis

I've been asked to show that the normals to a parameterized surface given by: $x(u,v) = (f(u)cosv,f(u)sinv,g(u)), f(u) \neq 0, g'(u) \neq 0$ all pass through the z-axis. I've computed the normal to ...
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1answer
20 views

Jacobi determinant for high-dimensional sphere inversion

I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the ...
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28 views

Preimage of a submanifold is a submanifold - Transversality

It is well known that if a smooth Map $f : M \to N$ between two smooth manifolds (finite dimensional) is transversal to a submanifold $L \subset N, L \pitchfork f$, than $f^{-1}(N)$ is a submanifold ...
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24 views

Show the first Chern class of a $U(1)$ bundle is integral.

I am working from John Baez's book: "Gauge Fields, Knots and Gravity". So I will stick to the notation used in that book. I am stuck at exercise 122 of part II (page 283), it reads: Show that if ...
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11 views

Bijective local isometry to global isometry

Suppose that I have a bijective local isometry $f: X \rightarrow Y$ where $X$ and $Y$ are length spaces. Can I show that $f$ is a global isometry? My thought is to consider a path $\gamma$ from $x$ to ...
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32 views

Smooth approximation to a continuous curve

Let $\gamma: [0,1] \rightarrow M$ be a continuous curve in a smooth manifold $M$. Is there a standard way to approximate $\gamma$ by a smooth curve? My thought was to look at every point $p$ where ...
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1answer
17 views

A certain zeta function; or, the determinant of the Laplacian plus a constant on the circle

I am interested in a certain "zeta function," a meromorphic function of $s \in \mathbb{C}$ that depends on a real parameter $\alpha \neq 0$. It's defined for the real part of $s$ large by $$ ...
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37 views

Is it possible to define submanifold like this

Wikipedia offers the following definition for an (embedded) submanifold: An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a ...
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22 views

An compute of Riemannian geometry [on hold]

According to Einstein summation convention , $g_{ij}$ is metric tensor,and $f$ is a real function. Show that : $$ g_{jl}g^{ij}\frac{\partial f}{\partial x^i}g^{kl}\frac{\partial f}{\partial ...
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20 views

Image is not a manifold when considered as a subset: how is this possible?

Wikipedia offers two definitions of a submanifold: One is that it is the image of an immersion. But I can't make sense of the remark that " in general this image will not be a submanifold as a ...
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34 views

Inverse Function Theorem: is this true?

The inverse function theorem is usually stated as follows: Let $f:\mathbb R^n \to \mathbb R^n$ be a smooth map and let $x_0$ be a point such that $\det J_f (x_0) \neq 0$. Then there exists an open ...
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1answer
24 views

Is $\omega = \theta d\theta + zdz$ one 1-form in $S^2$ with cylindrical coordinates?

Take the $S^2$ sphere with cylindrical coordinates. We now that $\alpha = d\theta\wedge dz$ is the symplectic form of this manifold, with $\theta \in [0,2\pi)$ and $z \in (-1,1)$ . Following the same ...
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58 views

What are the prerequisites for Michael Spivak's monumental A Comprehensive Introduction to Differential Geometry?

What are the prerequisites for Michael Spivak's monumental A Comprehensive Introduction to Differential Geometry? In particular for volume 1? Are these 5 volumes self-consistent in the sense that a ...
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1answer
35 views

Stuck with a problem of calculus of variation in the proof that a minimizing curve is a geodesic

I'm reading the proof of the proposition that states that every minimizing curve is a geodesics when it is given an unit speed parametrization. In the proof appears the following quantity : $$ ...
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2answers
33 views

What rhumb lines of a torus are periodic $C^1$ curves?

This is a question coming from an old (French) geometry book. Take a 3D torus. Study the rhumb lines of the torus and find the ones that are periodic $C^1$ curves. In particular, it is mentioned that ...
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27 views

Extending bounded smooth curve in $\mathbb{R}^n$ [on hold]

If I have a smooth curve $\gamma:(0,1]\rightarrow\mathbb{R}^n$ such that the image of $\gamma$ is bounded can I extend this curve so that it smoothly approximate another curve whose endpoint agrees ...
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Differential geometry qsn [on hold]

Find the curvature and torsion of the curves given by: $r = (a(3u-u^3),3au^2,a(3u+u^2))$ $r = a(1 + \cos u), a \sin u,2a \sin \frac u 2)$
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1answer
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Fitting an ellipse to a point with the first and second derivatives specified

I am trying to fit an ellipse to the end of a curve, $y(x)$, such that first and second derivatives of the curve, $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, are preserved at the contact point, $(x,y)$, ...
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1answer
20 views

Laplace-Beltrami operator as sum of orthogonal projections

Let $M$ be a submanifold of $\mathbb R^l$ with the induced metric. Let $(\xi_\alpha)$ be the standard orthonormal basis on $\mathbb R^l$. For each $x \in M$, let $P_\alpha(x)$ the projection of ...
2
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1answer
25 views

Evolution operator always in $SU(n)$?

Think about the evolution operator $U$ in Quantum Mechanics for finite dimensional systems. Then this operator satisfies an equation $$U'(t) = -iHU(t)..$$ Here, I assume that $H$ is ...
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2answers
51 views

Locus with line segments ratio constant.

$OAB$ is a rotating radial ray through origin $O$. Find a continuous curve through A and B so that quotient $OA/OB$ is constant, excluding Euclidean motion of rotation around $O$. A and B can also be ...
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3answers
126 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
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Uniqueness of covariant derivative in Do Carmo

2.2 Proposition: Let $M$ be a differentiable manifold with an affine connection $\nabla$. There exists a unique correspondence which associates to a vector field $V$ along the differentiable curve ...
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1answer
39 views

Vector field left-invariant then also its respective flow?

I was wondering whether left invariance of a vector field $X$ to a respective Lie group $G$ (so $dL(a)(x)(X(x))= X(ax))$ is transfered to the respective flow defined by $\frac{d}{dt} \phi^{t}(x) = ...
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43 views

Identity regarding the components of a dual basis

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ ...
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Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
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1answer
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Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where ...
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2answers
74 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
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Definition of $k$-precosymplectic manifold

A precosymplectic manifold of rank $2r$ is a triple $(M,\omega,\eta)$ where $M$ is a smooth manifold of dimension $2m+1$, $\omega$ is a closed 2-form on $M$ and $\eta$ is a closed 1-form on $M$ such ...
3
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0answers
41 views

some important proofs about adjoint operators [duplicate]

I was told that the formal adjoint of the gradient is the negative divergence. Let $A : H\to H$ be a bounded, linear operator, The adjoint of $A$, i.e. $A^*: H\to H$ satisfies \begin{equation*} ...
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Reference request: Tubular neighborhood theorem for non-closed manifolds via the exponential map.

Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and ...
2
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Non-hyperbolic zeros of vector field

I'm wondering the following: Let $V$ be a vector field on a (compact Riemannian) smooth manifold $M$ with non-degenerate zeros. Let $p$ be a non-hyperbolic zero of $V$. Can we perturb $V$ slightly so ...
3
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1answer
20 views

limiting tangent line is parallel to asymptotic line

For a (infinitely, if necessary) differentiable curve $$ A(t) = (x(t), y(t)) $$ which diverges at $t_0 \in [-\infty,\infty] $, that is $$ \lim_{t \to t_0 } | x(t)^2 + y(t)^2 | =\infty $$ if there is ...