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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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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18 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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8 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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16 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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An compute of Riemannian geometry

According to Einstein summation convention , $g_{ij}$ is metric tensor,and $f$ is a real function. Show that : $$ g_{il}g^{ij}\frac{\partial f}{\partial x^i}g^{kl}\frac{\partial f}{\partial ...
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12 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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28 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
17 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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1answer
45 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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24 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
27 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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24 views

Extending bounded smooth curve in $\mathbb{R}^n$

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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15 views

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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15 views

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
15 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 ...
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1answer
21 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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1answer
43 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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94 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
35 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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37 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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28 views

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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28 views

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
72 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 ...
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34 views

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 ...
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23 views

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
19 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 ...
3
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1answer
35 views

Pullback of a normal bundle

Consider $\Sigma$ a compact surface embedded into a compact 3-manifold, such that $\Sigma$ is diffeomorphic to $\mathbb{R}\mathbb{P}^2$ (real projective plane) and $\varphi:\mathbb{S}^2 \to \Sigma$ is ...
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31 views

On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
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1answer
36 views

Metric tensor for n-sphere in ambient coordinates

Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$: $$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$ What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ ...
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1answer
39 views

Ricci curvature along Killing vector field

If $V$ is a Killing vector field, I need to prove that $$V^{m}\nabla_{m}R = 0$$ where $R$ is the Ricci scalar $R = g^{mn}R_{mn}$. I´m having some trouble with this, I already showed that ...
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112 views

Closed form on any submanifold closed?

Let $M$ be a manifold and $\omega$ a closed differential form, so i.e. $d \omega =0.$ If I now consider a submanifold $N$ of $M$. Does this mean that $\omega$ is still closed on $N$? This statement ...
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59 views

Why is this not an inconsistency in elementary Lie theory?

I made an observation last week, and it has bothered me ever since. Recall the formulae ...
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22 views

Sending unitary group to reals, second differential at critical points.

Let$$X = U(n) = \{B \in \text{GL}_n(\mathbb{C}): BB^* = I\}$$be the group of $n \times n$ unitary matrices over $\mathbb{C}$, viewed as a real manifold. Fix a diagonal $n \times n$ matrix $A$ with ...
3
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1answer
43 views

Lie Derivative Equals to Lie Bracket

I am reading the book Introduction to Smooth Manifold written by M.Lee. I am confusing with the concept of Lie derivative. We have $\mathcal{L}_XY=[X,Y]$. However we have ...
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1answer
25 views

Linearization of divergence of a vector field?

Let's $X$ is a fixed smooth vector field on semi-Riemannian manifold $(M,g)$. For a symmetric 2-tensor field $s$, and for sufficiently small values of $t$, $\tilde{g}=g+ts$ is a semi_Riemannian metric ...
2
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1answer
48 views

Computing Rham Cohomology

Suppose that we have a $C^{\infty}$ manifold $X$ with and atlas $\mathcal{A}=$($U_{\alpha},\varphi_{\alpha}$) such that for every two intersecting open sets $U,V \in \mathcal{A}$ the intersection is ...
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2answers
34 views

Uniqueness of tangent plane

Let $\Sigma$ be a smooth surface defined as a surface admitting a parametrisation $\boldsymbol{r}:D\subset\mathbb{R}^2\to\mathbb{R}^3$ such that $\boldsymbol{r}$ is of class $C^1(\mathring{D})$ (and ...
3
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1answer
55 views

Is there a natural Riemannian structure on the total space of a vector bundle?

Suppose $B$ is a Riemannian manifold and $\pi: E \to B$ is a smooth vector bundle equipped with a metric. Is there a natural Riemannian metric on $E$, i.e. a bundle metric on $TE\to E$? It seems ...
3
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1answer
56 views

Why $\dfrac{\partial \sigma}{\partial u}=\dfrac{\partial \sigma}{\partial \bar{u}}$?

According to Elementary Differential Geometry by A N Pressley: (${\bf \sigma}_u=:\dfrac{\partial \sigma}{\partial u}$). The above text several times assuming that $\dfrac{\partial ...
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1answer
43 views

If two objects have the same gaussian curvature, are they the same up to isometries?

I was reading about Gauss Egregium Theorem but I'm not sure if I understand it well. Intuitively, what does it mean? It is true that if two objects have the same Gaussian curvature, then they are the ...
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Show that the outward unit normal is smooth vector field

Exercise 2.1.8 of Guillemin and Pollack asks us to prove that if $X^m \subset \mathbb{R}^n$ is an embedded submanifold with boundary, then the outward unit normal to $\partial X$ is a smooth function ...
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1answer
45 views

Symplectic form and wedge sum

The wedge sum of $k$ symplectic 2-forms is given by ( if $\omega = \sum_i e_i^* \wedge f_i^*$) $$ \omega^k = k! \sum_{1\le i_1 <...<i_k \le n} (e_{i_1}^* \wedge f_{i_1}^*) \wedge ... \wedge ...
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1answer
36 views

Is the canonical bundle topologically trivial?

Suppose $X$ is a $n$-complex dimensional complex manifold, we can form its canonical bundle $K_{X,\mathrm{hol}}=\bigwedge^n\Omega_{X,\mathbf{C}}$. Usually this bundle is not holomorphically trivial. ...
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1answer
42 views

A diffeomorphism which maps geodesics to geodesics preserves the connection?

Let $(M,\nabla^M),(N,\nabla^N)$ be two smooth manifolds with given (affine) connections on their (tangent bundles). We say a diffeomorphism ,$\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ is an ...
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1answer
51 views

Why derivation is a directional derivative?

Suppose $M$ is a smooth manifold, and $X\in T_pM$. Why for every derivation in $p\in M$ exist tangent vector $X\in T_pM$, witch satisfies $L_p(f) = X\cdot f$ for every smooth $f\colon M\to\mathbb{R}$? ...
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1answer
31 views

The complex structure of a complex torus

A complex torus $X$ of complex dimension $1$ is just a quotient of $\mathbb{C}$ by a lattice $\Lambda=\mathbb{Z}\oplus \tau\mathbb{Z}$ where $\tau=a+ib$ is a complex number with $b>0$. The complex ...
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2answers
67 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
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1answer
27 views

Prove that $\int_V (\mathbf{x} - \mathbf{x}_c) dx dy dz= 0$, $x_c$ is the centroid of the volume $V$

This is what I came up with and I am not sure if it is correct, and I would like to know if there is another, maybe purely geometrical, way of obtaining the equation. A centroid will be the center ...