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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Getting started with contact bundles

I'm currently reading William Burke's book Applied Differential Geometry and he uses a lot in the development of Lagrangian Mechanics the notion of a Contact Bundle. He does explain intuitively what ...
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Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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93 views

Consequences of tubular neighbourhood theorem

Consider an oriented manifold without boundary S embedded in an oriented manifold M. The tubular neighbourhood theorem says that there is a neighbourhood T of S in M which is diffeomorphic to ...
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58 views

Laplace-Beltrami operator for Kahler 2-form

Laplace-Beltrami operator for Kahler 2-form: $$\triangle\Omega(X,Y)=d\delta\Omega(X,Y)+\delta d\Omega(X,Y)$$ We know that $$\delta d\Omega(X,Y)=-\sum_{k}(\nabla_{e_{k}}d\Omega(e_{k},X,Y))$$ where ...
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Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
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51 views

Wedge product of a one- form and a Kähler form

Let $x$, $y$, $v$, $w$ be coordinates on $R^{4}$ and $g$ be the Riemannian metric whose matrix with respect to these coordinates is $$g=\left ( \begin{array} {cccc} 1 & 0 & -kx & 0\\ 0 ...
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58 views

Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some ...
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33 views

Proving a global property of the connection from a property of local connection matrices

Let $D$ be an $m$-dimensional distribution on an $n$-dimensional manifold $M$. Let $U\subset M$ be an arbitrary open subset such that on $U$ we can define vector fields $e_i$ such that for each $x\in ...
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50 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...
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56 views

Prove that a canonical bundle is trivial

Consider a function $f \in C^{\infty}(\mathbb{R}^n)$, $y \in Reg(f), M=f^{-1}(y)$. Prove that the canonical bundle of M is trivial. I have an hint but I don't know how to use it: consider the open ...
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75 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
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92 views

How to prove that the composite function is smooth

Let $f:M \to N$ and $g:N \to K$ be smooth functions, where $M,N$ and $K$ are smooth manifolds. How to prove that the composite function $g \circ f$ is smooth, noting that the Chain Rule only applies ...
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69 views

construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
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46 views

Some properties in differential geometry of curves and surfaces

Let $\beta, \alpha$ be curves in $\mathbb{R}^3$ parametrized by arc length. Suppose $\beta$ is obtained by rotating $\alpha.$ Let $t_{\alpha}, n_{\alpha}, b_{\alpha}$ (resp. $t_{\beta}, n_{\beta}, ...
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27 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
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57 views

Vector fields on homogeneous space $G/H$

I am trying to understand why the vector fields on $G/H$ are maps $X:G\rightarrow \frak{g}/\frak{h}$ satisfying $X(rh)=Ad^{-1}(h)X(r),\,\, h\in H.$ Any hint would be greatly appreciated!
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165 views

Some Advice on My Undergraduate Paper

My teacher wants me to read something about "Differential Geometry in $R^3$" and choose a topic as a paper. Now I have finished these books. And I am interested in some topics below: $(1)$ ...
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55 views

Doubt on the definition of topological manifold

I need some clarifications on topological manifolds. My professor has defined them as second countable, connected, Hausdorff topological spaces which are locally homeomorphic to $\mathbb{R}^n_+$, ...
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173 views

The Fubini-Study metric and its associated form.

Assume $ds^2$ is Fubini-Study metric, and $\omega = -\frac{1}{2}Im \{ds^2\}$ is its associated form. Let $V_n = \int_{\mathbb{C}P^n} \omega^n / n!$ the volume of $\mathbb{C}P^n$. Let $V \subset ...
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309 views

The curvature and torsion of the tangent indicatrix

Let $\alpha$ be a unit speed curve. Its tangent indicatrix $\sigma$ is defined by $\sigma(t)=T(t)$. Find torsion and curvature of $\sigma$ with respect to the torsion and curvature of $\alpha$. ...
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107 views

Expressions for exponential map and parallel transport

This came up in a paper I was perusing. The authors list three formulas which I have not been able to comprehend. Here M is supposed to be a simply connected space form. Then for the exponential map ...
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68 views

Newton polygon and asymptotic behavior near a singular point

As we know, Newton polygons could be used to determine the Puiseux series of algebraic curves (see, for example, Kirwan's Complex Algebraic Curves, chapter 7). Different branches correspond to ...
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139 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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92 views

The relation between conformally related metrics and conformal vector fields?

Two metrics $g_{1}$ and $g_{2}$ are conformally equivalent metrics if $g_{2}=e^{2\theta}g_{1}$ A vector field $X$ is called conformal if $L_{X}g=2\theta g$ where $L_{X}$ is the Lie derivative with ...
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What can we say about the integral curve of a vector field on the warped product manifold?

Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I ...
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44 views

comformal surface parameterization of simply-connected Riemann surface with boundary

$M$ and $S$ are two simply-connected surfaces with boundaries $\partial M =\partial S \neq \emptyset$, respectively. Both $M$ and $S$ are disk-like and smooth enough. Here, we assume $M$ is planar ...
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67 views

Frenet equations used in curved space

I've been reading this article by M. Abramowicz. Very interesting article but I can't make sense of one detail: the author writes Eq.$(9)$ stating that we can use (that) Frenet equation to define the ...
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24 views

Orientability of Ringed Space

Differential manifold can be defined in two ways. One definition is a topological space equipped with an atlas and transition maps. Another definition is a topological space equipped with a sheaf of ...
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42 views

Minimal immersion

Let $N$ be an $n$-dimensional manifold immersed in an $l$-dimensional manifold $L$ by immersion $\iota: N\to L$. And let $\iota$ be minimal immersion. On the other hand, let $\phi: M\to N$ be totally ...
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52 views

Smooth parametrizations of continuous curves

Consider the curve $t\to (t,|t|)$ in $\mathbb R^2$. Even if it has a cusp in $(0,0)$ i can reparametrize it with a smooth function. Take for example $$ t\mapsto \begin{cases}(\mathrm e^{-1/t},\mathrm ...
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67 views

Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
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59 views

Tangent space of Sym(n,$\mathbb{R}$)

I want to compute the tangent space of the space of symmetric real matrices Sym(n,$\mathbb{R}$) = $\{A\in GL(n,\mathbb{R})|A^t=A\}$. There are two different (and seemingly contradictory) statements ...
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86 views

Identites of Riemann curvature tensor on orthonormal frame

Suppose we consider a Riemannian manifold and a local orthonormal frame $\{Y_i\}$. I was wondering whether, for the Riemann curvature tensor $R$, there are identities with regards to expressions of ...
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Determine the number of revolutions the normal to a curve makes as it moves along a curve in three dimensions?

Given a point on a 3D curve, how many full revolutions does the normal to the curve at the point make as the point moves over the curve? Assume the point stops when it reaches the place where it ...
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45 views

immersions of spheres with handles minus disks

i am at a loss as far how exactly to immerse a sphere with genus g minus a disk into the plane. Thanks in advance
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48 views

homeomorphism still isotopic to the identity after the deletion of points.

I have a surface $M$ (without boundary) and a homeomorphism $f:M \rightarrow M$, which is isotopic to the identity on $M$. If I delete two points $x$ and $f(x)$ from the surface, I get a ...
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30 views

Obstruction of such gauge choice

Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard ...
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38 views

Orientability of smooth structures on the real line

How I can prove that every every smooth structure on the real line (with the standard topology) is orientable?
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59 views

prove that the shapes are isometric

I want to prove that the shapes are isometric. How to prove? There is no info except for the picture. First of all I need to write surface patches. Please can someone help me? The definition of ...
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32 views

Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
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146 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
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54 views

Differentiating the exponential map of a unitary group

I consider a unitary group $U \in U(N)$ with an exponential map: $ U = \exp (iH)$, with $H$ hermitian. I am not sure if I can do the following calculation: $U^\dagger d U = \exp (-iH) i dH \exp (iH) ...
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43 views

Holder estimate for eigenfunctions of Laplace operator on sphere?

I would like to ask if there is a holder esitimate for eigenfunctions of Laplace operator on sphere? I mean the esimate for \begin{equation} \|\partial^{\alpha}h_n\|_{L^{\infty}(S^{d-1})} ...
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show that $g\circ f :S_1 \to S_3$ be also smooth and $d_p(g\circ f)=(d_{f(p)}g)\circ (d_pf)$

proposition: Let $S_1$, $S_2$, $S_3$ be 3-regular surfaces and $f:S_1 \to S_2$ and $g: S_2 \to S_3$ be smooth maps. Then show that $g\circ f :S_1 \to S_3$ be also smooth and $$d_p(g\circ ...
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117 views

$k$-jets of sections of a vector bundle..

I need some help for establishing a connection between two definitions of $k$-jets: Algebraic Definition: Let $E\rightarrow M$ be a smooth vector bundle and define the ideal of $\Gamma(E)$: ...
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56 views

The local chart of the embedded submanifold

Let $M$ be a $m$-dimensional smooth manifolds with boundary. $N$ is an embedded submanifold of $M$ such that $\partial N = \partial M \cap N$ and $N$ is transverse to $\partial M$, that is, for any $x ...
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50 views

Relative de Rahm cohomology computation for two disjoint circles embedded in R^2

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
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94 views

Geodesics on pseudo-Riemannian manifolds

Consider a Riemannian manifold $M$, with a metric $g$. We can find univocally the Levi Civita connection $\nabla$ on $M$, and so a covariant derivative $D_t$ (associated to $\nabla$) along curves. A ...
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42 views

Euler characteristic in 4 dimension

Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as: $E_4 = \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$ with $R^{ab}$ is the curvature 2-form. Perturb the ...
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37 views

$\mathbb{RP}^3 \rightarrow \mathbb{CP}^1$ defines a principal U(1)-bundle

I have to show that the map $\pi: (x_o : x_1 : x_2 : x_3) \in \mathbb{RP}^3 \rightarrow (x_0 + i x_1) : (x_2+ ix_3) \in \mathbb{CP}^1$ defines a principal U(1)-bundle. The two standard coordinate ...