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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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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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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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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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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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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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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1answer
18 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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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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2answers
65 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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29 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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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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1answer
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What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
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1answer
337 views

Riemann tensor symmetries

The Riemann tensor has its component expression: ...
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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 ...
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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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29 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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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 ...
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+100

Newton iteration on Riemannian manifolds

Suppose $f:M \to N$ is a smooth map between complete Riemannian manifolds of the same dimension. Suppose $Df(m_0)$ is invertible, and $n$ is a point close to $f(m_0)$. Can we perform Newton iteration ...
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1answer
24 views

Representing a vector field locally

A couple of questions occured to me when reading about the Poincare-Hopf theorem. Any pointers or comments would be appreciated! Let $M$ be a closed oriented Riemannian manifold and $V$ a vector ...
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1answer
38 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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2answers
30 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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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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Lie algebra of R^n

Until now the only example of lie groups I have seen are subgroups of $GL_n$. Today I had the idea, that also $G=(\mathbb R^n,+)$ must be a lie group ($(\mathbb R^n,+)$ is a group with the ...
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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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21 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 ...
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2answers
516 views

Hodge double star operator

I want to prove that $**\omega=\left(-1\right)^{k\left(n-k\right)}\omega$, where $*$ is the Hodge star operator acting on differential $k$-forms $\omega$ on $\mathbb{R}^n$. Where can I find the proof ...
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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
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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
47 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 ...
4
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113 views

Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...
3
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1answer
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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 ...
3
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1answer
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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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3answers
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Preservation of the curvature tensor implies preservation of the connection?

For every connection on a smooth manifold there is a corresponding curvature tensor. Any diffeomorphism $\phi:(M,\nabla^M)\rightarrow(N,\nabla^N)$ which preserves the connection (in the sense of ...
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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 ...
2
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1answer
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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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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
15 views

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
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
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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Proof of Brouwer fixed point theorem using Stokes's theorem

$\omega$ is the volume form on the boundary B -ball $f\colon B \to \partial B$ $$ 0 < \int_{\partial B}\omega = \int_{\partial B} f^*(\omega) = \int_{B} df^*(\omega) = \int_B f^*(d\omega) = ...
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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
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Proving a given set is a submanifold

Let $S \subseteq \mathbb R^n$. I have been faced with showing that $S$ is a submanifold and I have some ideas but I want to get the complete picture. (Main) Question 1: What methods are there to ...
3
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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
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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 ...
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Proof of co-area formula in terms of level set function

How to prove the following theorem, Let for each $t \in [0,T]$, $\phi( t,\cdot) : {\bar \Omega} \rightarrow R$ be Lipschitz continuous and assume that for each $r \in ( \textrm{inf}_{\Omega} \phi, ...
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18 views

Diffeomorphism and Orientable double cover

Suppose that the orientable double cover of two homeomorphic surfaces are diffeomorphic, is it true that these surfaces are diffeomorphic?
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Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...