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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53 views

Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov

Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
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73 views

Smoothing corners of a handle attachment

Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
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228 views

Difference between parallel transport and derivative of the exponential map

Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then we have to ways to map $T_pM$ to $T_{c(t_0)} ...
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111 views

Differential geometry textbook or lecture notes on the riccati equation and riccati inequality

I took a course on differential geometry and didn't get one specific topic well, so I am searching on some additional metrial to understand it in a better way. This wasn't a course about classical ...
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89 views

How to construct an orthogonal coordinate system from a smooth planar curve?

Given a planar curve $$\gamma:\mathbb R\to\mathbb R^2, t\mapsto \gamma(t) \text{, normalized to } |\gamma'(t)|\equiv1,$$ the tangential vector $$T(t) = \gamma'(t)$$ and the normal vector $$N(t) = ...
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82 views

Why does the map $x^2$ have constant rank?

I'm just trying to wrap my head around the rank of a map via some examples. Now, if I have the smooth map of manifolds $F:\mathbb{R} \to \mathbb{R}, F(p) = p^2$, then the differential is given by ...
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159 views

Difference between distance between two points and metric

if i have a line element given e.g. $ds^2=\frac{dx^2+dy^2}{2y} $ is it then always possible to derive a distance between two points in this metric? and how would one determine the length of a curve if ...
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33 views

How is integration of differential form defined as, and how to calculate it

How is integration of differential form defined as? And how does one calculate the value of integration?
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55 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
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82 views

Volume form on complex space, notation

If $\Omega = dz^{1}\wedge..\wedge dz^{n}$, $z^{1}$,..,$z^{n}$ are complex numbers. What does the notation $\Omega (x^{1}\wedge.. \wedge x^{n})$ means? $x^{1},..,x^{n}$ are real parts of ...
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104 views

Immersed Surfaces in Hyperbolic Space with Positive Intrinsic Curvature

Does anyone know of an example of a noncompact, immersed surface in hyperbolic 3-space with positive intrinsic curvature that is NOT embedded?
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72 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
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33 views

Nearly Kaehler and special Kaehler manifolds

We know that the most important example of a nearly Kaehler manifold is the sphere $S^{6}$ and that $(\nabla_{X}J)Y=-(\nabla_{Y}J)X$ is valid in this case (J - an almost complex structure). Similar ...
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340 views

How to derive covariant derivative and Lie derivative of tensors

1) As title says, how does one derive the following equation for covariant derivate of tensor: $A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$ where ...
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33 views

Dual connections, examples

If $\overline\nabla^{*} $ is a dual connection of connection $\overline\nabla $, and we have the Gauss equations: \begin{align} \notag \overline\nabla^{*}_{X}Y=\nabla^{*}_{ X}Y + h^{*}(X,Y) ...
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34 views

Boundaries- regularity and local parametrization

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...
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24 views

density of $\mathcal{C}_1$ surface in a point

Let us have an $A \subset \mathbb{R^d}$, that is a $k$-dimensional $\mathcal{C}_1$ surface (obviously $k<d$) and let $a \in A$. Why then is $\Theta^k(A,a)=1?$ Of course $\Theta^k := \lim_{r ...
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240 views

jacobian involving SO(3) exponential map: $\log(R \exp(m))$

I would like to compute the 3 × 3 Jacobian of $$ \log(R \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...
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133 views

De Rham cohomology of $\mathbb R^3$ without lines and a circumference

I am trying to calculate De Rham cohomology of the following spaces: $X=\mathbb R^3\setminus r$ where $r$ is a line; $Y=\mathbb R^3\setminus (r \cup \gamma)$ where $r$ is a line and $\gamma$ is a ...
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75 views

Sufficient conditions for Hessian definiteness for critical points of functionals

Let $C$ be the set of smooth curves from the unit interval into $\mathbb{R}^n$. Let $f : C \rightarrow \mathbb{R}$ be a functional on these curves given by $f(x) := \int_0^1 L(x,\dot{x}) dt$. Define ...
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99 views

Prove $\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$

$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$ The above is an identity frequently used in ...
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40 views

Approximation/Representation of local stable manifolds

I will give two preceding theorems and the question, which uses both, follows afterwards: Let $M$ be a smooth compact Riemannian manifold of dimension $n$ with a smooth measure $\mu$. $T_{x}M = ...
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74 views

Ricci tensor in complex space forms

Let \begin{align} \notag f:M^{2n}\to CQ_{c}^{N} \end{align} be an isometric immersion of a Kaehler manifold into a complex space form. We consider an orthonormal basis $Y=X_{1},..,X_{2n}$ and then ...
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130 views

Wedge Product Vocabulary

I am looking for a word or phrase that is similar in meaning to the "support of a function", but in the context of differential forms. The "support of a function" is the subset of the domain of a ...
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114 views

Gaussian curvature in terms of orthonormal basis

I need to prove the following formula for Gaussian curvature $K$ of an open subset $V$ of a surface $S$ (and then use it to find the Gaussian curvature of an ellipsoid): $$ K=\frac{\big<D(fN)(v_1) ...
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103 views

Computing evolute o f a curve by finding the surronding of the normal rects.

So, I have to compute the evolute of the curve: $y^{2} = 2px$ But I have to do that by computing the surronding of the normal rect family. So I start taking the positive side of the function and ...
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38 views

if the equality $a^i_ju_i=Ku_j$ holds for any covariant vector $u_i$ such that $u_iv^i=0$ , show that $a^i_j=K\delta^{i}_{j} +p_jv^i$

if the equality $a^i_ju_i=Ku_j$ holds for any covariant vector $u_i$ such that $u_iv^i=0$ where $v^i$ is a given contravariant vector, show that $a^i_j=K\delta^{i}_{j} +p_jv^i$. i am completely stuck ...
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41 views

Functionals defined on curves

I am looking for classical (real) functionals defined on rectifiable curves (neither necessarily simple nor closed) in either $\mathbb{R}^2$ or $\mathbb{R}^3$. There's length, turning number, total ...
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103 views

Implicit function theorem example of three variable

Let $\displaystyle \phi(x,y,z)=x^{2}+4y^{2}-2yz-z^{2}$ and let $\displaystyle x_{0}=2e_{1}+e_{2}-4e_{3}$. So i have to verify the hypotheses of the implicit function theorem for the above example. I ...
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113 views

group action - compact complex torus with $H^2_{DR}(X,C) = 0$ (de Rham cohomology)

If $\mathbb{Z}$ acts on $\mathbb{C}^n \backslash \{0\}$ by $(m,z) \mapsto 2^m\,z$, I need to show that $H^2_{DR}(X=(\mathbb{C}^n \backslash \{0\})/\mathbb{Z},\mathbb{C})=0$. I start with showing it ...
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69 views

What is the diameter of a manifold?

From wikipedia the definition of diameter is the supremum of the distance function of the set. But what if there is no obvious distance function, say for the set $SO(n)$. Also how does this work when ...
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70 views

how to calculate the derivative of a plane wave in non commutative geometry

Shahn Majid and Eliezer Batista find the derivative of a wave plane in their paper: Non Commutative Geometry of Angular Momentum Space $U \mathfrak{su}(2)$. They obtained the non commutating ...
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119 views

Generalization of Gaussian Curvature?

Consider a 2 dimensional manifold M parametrized by coordinates (x,y) embedded in $\mathbb{R}^{3}$. Thee is a smooth curve in the manifold given by $(\gamma_{1}(t),\gamma_{2}(t))$ with ...
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164 views

General definition of a line

In the book on Linear Algebra that I am using, the author defines a line in an arbitrary vector space $V$, given direction $ 0 \neq d \in V $ and passing through $ p \in V$ as $ l(p;d)= \lbrace v\in ...
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104 views

Characterization of bounded geometry - Reference-request

I'm looking for a proof of an equivalence that can e.g. be found in a paper by Shubin 'Spectral theory of elliptic operators on non-compact manifolds' (Appendix A.1.1 below Def. 1.1). It's about ...
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73 views

Is this mean curvature?

Suppose $N_t:=\partial B(p, t)\subset M^{n+1}$ be the distance sphere in a Riemannian manifold. Let $\{x_1, \cdots, x_n\}$ be a coordinate of the distance sphere $\partial B(p, t)$. Hence $\{x_1, ...
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113 views

Difference between two alternative definitions of a differentiable manifold.

Two or more sources of study I use on Differential Geometry,most notably, Barrett O'Neill's book, defines an $ n $ dimensional manifold as follows: A set furnishes with a collection $ P $ of abstract ...
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116 views

Is the following method of determining a curve from its normal vector valid?

One of my homework problems asks us to show that the curvature and the torsion of a regular parameterised curve with non-zero torsion everywhere are uniquely determined, when we already have the ...
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221 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
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66 views

show that subset of complex projective space is a submanifold

Let n, m $\in \mathbb{N}$. I'm trying to show that $M(n,m) = \{[z_0 : z_1 : … : z_n] \in \mathbb{C}P^n | \sum^{n}_{i=0} z^m_i = 0\}$ is a submanifold and codim(M(n,m))=2. My idea was to use ...
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52 views

Prove that $X$ parametrizes a regular surface $M$ in $\mathbb{R}^3$ and determine for which values $p$ the curve $y$ is geodesic on $M$.

Real valued functions $f,g:\mathbb{R}_+ \rightarrow \mathbb{R}$ are $f(u) = e^{-u}$ and $g(u) = \int \sqrt{(1-e^{-2t}} dt$. Given $X:\mathbb{R}_+ \rightarrow \mathbb{R}^2$ with $X(u,v) = [f(u) \cos ...
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67 views

Isomorphism between tensor product and set of all tensors

I'm new to tensor and quite confused. First of all, could anyone provide any friendly reference about tensors which explains the idea behind those boring-looking definition? Then is my problem. Let ...
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81 views

Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange ...
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74 views

Problems about dual map, cotangent bundle.

I have no idea what dual space and dual map are, so have much trouble understanding cotangent bundle when reading Lee's book. First of all, can anyone give me a introduction what the dual map and ...
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279 views

Extension of a smooth function on a set of manifold

I encountered the following proposition: If a function is smooth on an arbitrary set $S\in M$, where $M$ is a smooth manifold, then it has a smooth extension to an open set containing $S$. It seems ...
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237 views

What is a direct proof of isoperimetric inequality?

What is a direct proof of isoperimetric inequality? In another word, i am asking for a proof that the circle has the maximum area compare to other geometric shape with the same circumstance but ...
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97 views

Lie group GL(n,R) and the determinant map

Let $d : GL(n,R) \rightarrow R$ be the determinant map. I don't know how to prove that if the map $d* : T_{I_n}GL(n,R) \rightarrow T_1R$ is surjective, then the map $d* : T_AGL(n,R) \rightarrow T_1R$ ...
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97 views

Submanifold of $\mathbb{R}^4$

In the space of $2\times 2$ matrices, find explicitly the sets of matrices with 1)a single zero eigenvalue, 2) a pair of pure imaginary eigenvalues. Show that each set is a submanifold of ...
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31 views

Matrices of forms seen as sections of a vector bundle

Given a vector bundle $\pi: E \rightarrow M$ , what are the sections of $\Omega^p(\operatorname{End} E)$? are they just matrices whose entries are $p$-forms?
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100 views

What is the geodesic equation on $\mathbb{S}^{n}$?

Suppose $\gamma: \mathbb{R}\rightarrow \mathbb{S}^{n}$ is a smooth curve. Let $\gamma(t)=(x^{1}(t)...x^{n+1}(t))$. Let $\mathbb{D}^{n}$ be embedded into $\mathbb{R}^{n+1}$ by viewing ...