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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Question about index notation on partial derivatives.

I've been studying quantum field theory a little bit and I've encountered a notation like the following: $$\mathcal{D}_{x,x'}=\frac{\partial}{\partial x^\mu}\frac{\partial}{\partial ...
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rational first chern class of algebraic variety with zero kodaira dimension

Let $X$ be a compact kahler algebraic variety which has zero Kodaira dimension.Then integral first chern class vanishes?, what about rational first chern class?
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A question in algebraic variety in MO

I posted a question in algebraic geometry here http://mathoverflow.net/questions/178886/first-chern-class-of-fibers-of-compact-kaehler-algebraic-variety I need more detail on the answer Misha, I ...
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1answer
22 views

Properties of Tangent Vector of a Differentiable Simple Closed Curve in 2D

I think of a theorem about a differentiable simple closed curve in 2D that I would like to prove. Here it is: Let $C$ be a differentiable, regular, simple, closed curve in $\mathbb{R}^2$ parametrized ...
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20 views

Semi-Riemannian Manifold

I don't understand the principal idea of Semi-Riemannian Manifold. Why, if I have a metric tensor g on a smooth manifold M that is a symmetric nondegenerate (0, 2) tensor field on M of constant ...
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Given an $(n-1)$-form $\varphi$ on a smooth orientable $n$-manifold, there is a vector field $v$ such that $i_v\varphi = 0$.

I am working on the following problem. Let $M$ be a smooth orientable $n$-manifold, $n \geq 2$, and let $\varphi$ be a smooth $(n-1)$-form on $M$. Show that there is a vector field $v$ on $M$ such ...
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Bogus proof that the Liouville Form on the cotangent bundle is nondegenerate.

Suppose we have a manifold $M$ of dimension $n$ and its cotangent bundle $T^*M$. The Liouville form $\lambda$ on $T^*M$ is defined as $\lambda_{\omega_p} = \pi^*(\omega_p)$ where $\pi$ is the standard ...
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19 views

2-form corresponding to a contravariant vector and pseudo-forms

In Frankel's book he writes that in $R^{3}$ with cartesian coordinates, you can always associate to a vector $\vec{v}$ a 1-form $v^{1}dx^{1}+v^{2}dx^{2}+v^{3}dx^{3}$ and a two form $v^{1}dx^{2}\wedge ...
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Proving that homotopic maps have the same degree

Let $M, N$ be compact, connected, oriented manifolds. The degree of a map $f:M \rightarrow N$ is defined as the integer $k$ which satisfies $\int_{M} f^{*}\omega = k\int_{N}\omega$. Using the fact ...
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Why is the integral of any orientation form over $\mathbb{S}^1$ non zero?

I am trying to understand the proof of Theorem 17.21 in Lee's Introduction to smooth manifolds; however I am finding myself stuck right at the beginning. The statement I am having trouble with is: ...
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34 views

surface in $R^3$ that has $ds^2 = du^2/v^2 + dv^2/v^2$

For a 2D surface, if we have the first fundamental form of $$ ds^2 = du^2/v^2 + dv^2/v^2$$, can we integrate it out to get the parameter form of the surface embedded in $R^3$? I tried something like ...
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35 views

Is stereographic projection the only way to make a bijection between plane and sphere?

At a math exhibition, I learned the concept of stereographic projection for the first time. However, I am curious about the purpose of the stereographcal projection. I've learned that an area of ...
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1answer
21 views

Transversality question

I'm trying to solve this innocent problem. Let $X,Y\subset \mathbb{R}^3$ be two 1-dimensional manifolds. Show that there exists $v\in \mathbb{R}^3$ such that $X$ and $Y+v$ are disjoint. I know how ...
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1answer
20 views

Smooth map on differential manifolds

given two differential manifolds $M_1$ and $M_2$. I have to show that the projection $\pi: M_1 \times M_2 \to M_1$ is smooth. By definition, I then need to show that for a point $(a,b)\in M_1\times ...
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24 views

What is the hyperbolic plane equivalent to translation in euclidean space

in euclidean plane one can move polygons like rectangles, triangles etc. around by isometries, e.g. translations. For instance if we consider a rectangle with midpoint $0\in\mathbb{R}²$ then the image ...
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How does a differential act when we identify $T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N$?

It's fairly common to identify the tangent space of a product manifold as $$ T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N $$ where $p=(p_1,p_2)$, and the actual isomorphism is given by $v\in ...
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Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
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Differentiability of Quotient Maps; Open Books .

I would appreciate your comments re the differentiability of a quotient map $q$: Say I have a quotient manifold $(S\times I )/q ;I=[0,1]$ , where $S$ is a surface with non-empty boundary, where ...
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22 views

Gauss-Bonnet on negative surface

How is Gauss-Bonnnet theorem verified on a pseudosphere between cuspidal equator and its far-off centre on its symmetry axis? Should integral kg ds be zero in the limit at cusp of horn as a limiting ...
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Monotonic curvature and self intersections.

I'm trying to prove that if $\alpha: I \to \Bbb R^2$ is a differentiable curve, where $I$ is an interval, has strictly monotonic curvature, then $\alpha$ has no self-intersections. My attempt: We ...
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Dirichlet Eigenvalues of Laplace-Beltrami operator in Hyperbolic space

Consider the hyperbolic half-plane $\mathbb{H}=\lbrace (x,y)\in\mathbb{R}^2: y>0 \rbrace$ with standard Riemannian metric. The Laplace-Beltrami Operator can be written as ...
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1D manifold is diffeomorphic to $\mathbb R$ or to $S^1$

In his ODE classic V.I. Arnold considers easy to see (легко видеть) that every one-dimensional (connected and without boundary) differentiable manifold is either diffeomorphic to $\mathbb R$ (if it is ...
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differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
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1answer
27 views

Relation between geodesics and exponential map for Lie groups

I've been trying to find a clear explanation on the Internet but failed unfortunately, so I'm asking here. How does the exponential map relate to parallel transport and geodesics for Lie groups. If it ...
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1answer
55 views

If $S\subseteq M\times N$ is embedded, and $S$ and $\{p\}\times N$ intersect transversely in one point, then $\pi_M|_S$ is a diffeomorphism?

I'm trying to prove the equivalence of the following statements: Suppose $M^m$ and $N^n$ are smooth manifolds, $S\subseteq M\times N$ immersed, and $\pi_M$ and $\pi_N$ the projection maps. TFAE: ...
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Transition Functions for Cartesian Coordinate Systems

This is my first time using Mathematics SE (I've only used Physics and Astronomy before), so I apologize if this question is awkwardly phrased or incorrectly presented. I welcome any and all edits and ...
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$\Delta \vec{v}=0$ implies $\nabla\cdot \vec{v}=\nabla\times \vec{v}=0$?

\begin{align} \Delta\overrightarrow{v}&=\nabla(\nabla\cdot\overrightarrow{v})-\nabla\times(\nabla\times\overrightarrow{v})\\ ...
3
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Number of limit cycles: Counterexample of the extended Bendixson-Dulac criterion?

The problem concerns the number of limit cycles in the vector field of coupled differential equations (ODEs) in two dimensions, i.e. $$ \ \dot{x} = X(x,y)\\ \dot{y} = Y(x,y) $$ Specifically, let $$ \ ...
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Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
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Solution of eikonal equation is locally the distance from a hypersurface, up to a constant

Consider the Eikonal equation (with right handside 1) $$\sum_{i=1}^{n}(\frac{\partial u}{\partial x_i})^2=1$$ I want to see why any solution to this is locally the sum of a distance function from a ...
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Reference Request for differential ideals of Pfaffian forms on jet bundles

My setting is the following: Given two families of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1...n-m$), define $$\eta^i := \sum_{j=1}^m a^i_j\omega^j + ...
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The curvature of a Cycloid at its cusps.

My lecturer proposed a question to particular result regarding the curvature of a Cycloid (generated by circle of radius 1) at its cusps. Having left it as an open problem, I thought it'd be ...
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Given a measurable vector field, construct another such that together they form a basis at every point

Let $v_1:(0,1)\rightarrow \mathbb{R}^2$ a measurable function such that $v_1(x)\neq 0$ for all $x$. I wonder if it is possible to construct a measurable function $v_2:(0,1)\rightarrow \mathbb{R}^2$ ...
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1answer
50 views

Definition of a parallelizable manifold

My text that I am self studying from says that a manifold $M$ is parallelizable if it has a trivial tangent bundle which means that there is an isomorphism $\varphi:M\times \mathbb{R}^n\rightarrow ...
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1answer
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Proof, that equation decribes trace of curve, which is supposed to be simple

The equation, representing the trace of the curve $$ \varphi(x) = (\cos^3(t), \sin^3(t)) $$ is $1 = x^{\frac{2}{3}} + y^{\frac{2}{3}}$. Proof: Let $(x,y) = (\cos^3 t, \sin^3 t)$, then $x^{1/3} = ...
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Ways to represent functions of bounded variation as the difference of two monotone functions

I learnt (it is called Jordan's representation theorem) that every function $f : [a,b] \to \mathbb R$ with bounded variation could be represented as the difference $f = g - h$ of two monotone ...
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Finding the geographical coordinates

I have two circles $C_1$ and $C_2$ on the surface of the earth (sphere) intersecting at geographical coordinates $A$ and $B$ and also center of $C_1$ lies in $C_2$ and vice versa. I want to find the ...
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Compute the isotropy representation

Suppose $SU(1,1)$ acts on the open unit disc $\mathbb{D}$ in the natural way, by linear fractional transformations. The isotropy group is $U(1),$ since it stabilizes the point $0.$ I am trying to ...
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2answers
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Intuition for the compactness of real projective space $\mathbb{R}\mathbb{P}^n$.

I want to have an intuition for why the $n$-dimensional real projective space defined as $$\mathbb{R}\mathbb{P}^n:=\mbox{set of 1-dimensional subspaces of }\mathbb{R}^{n+1}$$ is compact. I don't see ...
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2answers
57 views

Explanation of non-orientability of the Möbius band

I have read about the orientation of manifold in the Tu's book. The book is very readable but the first example about non-orientable manifold is seemly hard to understand. On page 208, he gave an ...
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Modern mathematics for dummies

I have poor university mathematical education, but Math fascinates me, so I decided to educate myself for a bit. I know there is a dozen modern mathematical fields I know nearly nothing about like ...
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Constrained optimization using a cutting plane on a tetrahedron

Consider the figure below where $(a,b,c,d)$ is a tetrahedron and $p=(1-t)a+tb$ is a point on the $ab$ segment. If $n_a$ and $n_b$ are two unit vectors associated with $a$ and $b$, respectively, then ...
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64 views

Compute the derivative of Plucker Embedding

Let V be an n-dimensional vector space over $\mathbb{R}$, and $$\Psi: G(k,V)\rightarrow \mathbb{P}(\Lambda^k V)$$ be the Plucker embedding, where $$L=span \{u_1, ..., u_k\} \mapsto \Psi(L)=[u_1 ...
2
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1answer
37 views

Differential Form Over $S^2$

I was checking problems on differential forms and I found the following one. Consider the sphere $S^2 \subseteq R^3$ and the map $\omega_p : T_pS^2 \times T_pS^2 \rightarrow \mathbb{R}$ given by ...
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1answer
75 views

Is this a manifold?

I am trying to get started with differential geometry, and am having a difficult time wrapping my head around the concept of a manifold. One thing that would make it easier to understand would be if ...
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21 views

How to calculate the Maurer-Cartan form in the adjoint representation?

While I am reading a paper, I come across a difficulty. Here, we have a Lie group and we know its Lie algebra defined as $[G_a,G_b]=f_{ab}^{\phantom{ab}c}G_c$ with $G_a\in\mathfrak g$. Under the ...
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1answer
45 views

Definition of Null Hypersurface

I am a physics student confused with the notion of null hypersurface, so sorry if this question is very simple. Given a manifold $M$ and a hypersurface $H$ defined on it, we can always take the ...
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1answer
51 views

If $b$ is a regular value of $f$, $f^{-1}(-\infty,b]$ is a regular domain?

I'm trying to prove the first part of Proposition 5.47 of Lee's Smooth Manifolds, which is left to the reader. It says Suppose $M^m$ is a smooth manifold, and $f\colon M\to\mathbb{R}$ smooth. For ...
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1answer
33 views

Show that a set is a smooth curve and find a parameterization for it.

Let $S = \{ (x,y,z) \in \mathbb{R}^3 \mid x - yz + z^3 = 0 \}$. Let $\pi: \mathbb{R}^3 \to \mathbb{R}^2$ be such that $\pi(x,y,z) = (x,y)$. Let $H = \{p \in S \mid \pi_{\mid S}: S \to \mathbb{R}^2 ...
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Soft question:what are the prerequisites for this book?

I got an advice to read the book 《Topological Methods in Algebraic Geometry》 written by Hirzebruch,but I am not sure whether it's appropriate for me at this time. Have anyone ever read this book?What ...