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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Height function of a hypersurface

I was reading an article by do Carmo and Warner, which says: "By the height function for an oriented hypersurface at a point $p$ we shall mean the function defined on a neighborhood of the origin in ...
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30 views

integral of a vector field in $\mathbb{R}^n$

I'm wondering the definition of the integral of a vector field on a hypersurface in R^n. Here is what I guess, but I did not found it on the internet. Let $v$ be a vector field on $\mathbb{R}^n$ and ...
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28 views

Sobolev spaces on Riemannian manifold

I know one can define the Sobolev spaces on a Riemannian manifold as completions, but is there an equivalent definition that uses weak derivatives, like in the case of open sets in $\mathbb{R}^n$ ? ...
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24 views

Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general, continuously deforming manifolds?

I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or ...
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Connected components of the complement of a geodesic

I came across the book "Knots, Molecules, and the Universe: An Introduction to Topology" by E. Flapan which is quite nice. In the first chapters it is discussed how one can distinguish the sphere ...
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A piecewise regular simple closed curve bisects the area of the unit sphere if and only if it has total geodesic curvature 0

How can I prove that "A piecewise regular simple closed curve bisects (this curve splits the unit sphere into two pieces, the area of which are equal) the area of the unit sphere if and only if it has ...
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21 views

Gaussian curvature in polar coordinates

Find the expression for the Gauss curvature in the polar coordinates associated to the exponential map. I thought about using Gauss's lemma: if $(r,\theta)$ are polar coordinates in the tangent plane ...
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35 views

How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$?

$\nabla$ is Riemann connection and $R_{ij}=g^{kl}R_{ikjl}$. How to show $\nabla_a\nabla_c R_{ad}-\nabla_c \nabla _a R_{ad}=-R_{cade} R_{ae}-R_{ce} R _{de}$ ? Or generate commutator of generate ...
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Why do we need the $S \otimes L^{1/2}$ bundle product to determine a $Spin_c$ structure?

I am reading Marino's book on topological field theory and 4-manifolds and I am very confused in the construction of the $Spin_c$ structures for manifolds that do not admit a $Spin$ structure. In the ...
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18 views

Construct a lifting from the the total space of a fiber bundle to its spoace of 1-jets

I am reading the book "Convex Integrtion Theory by D.Spring" and need some help. Let $\pi:X \rightarrow V$ be a smooth fiber bundle over a smooth base manifold $V$. Let $h\in\Gamma^{0}(X)$ and ...
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22 views

finding geodesics on the surface $z=x^2$

Find all the geodesics on the surface $z=x^2$. I found the metric and the Christoffel symbols but i do not know what to do next, any hint ?
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35 views

symmetric and alternating tensors in differential geometry

The following is an excerpt from Chern's Lectures on Differential Geometry: I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the ...
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19 views

In proof of tangent space being a plane

can someone explain why this is true? If $\sigma$ is a surface patch of a surface $S$ and $p$ is a point on the image of $\sigma$ and if $p$ lies in the image of a curve $\gamma$ contained in $S$ say ...
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13 views

Property on the derivative of a wedge product of two n-forms

I'm trying to prove the following property of $n$-forms. When $w_1$ is a $n_1$-form on $M$, $w_2$ a $n_2$-form also on $M$, and $d$ denotes the exterior derivative $$\require{cancel} d(w_1\wedge ...
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21 views

Dirichlet problem for a ball in a Riemannian Manifold

I'm trying to work through the 2nd chapter "Volume comparison" of Jeff Cheeger's "Degeneration of Riemannian Metrics under Ricci Curvature Bounds". But with the last section I have some problems. ...
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Confusion regarding terminology in Pressley's E.D.G

Here are two definitions taken from page $77$ of Pressley's Elementary Differential Geometry - $2$nd edition. Definition $4.2.1$ A surface patch $\sigma: U \to \Bbb R^3$ is called ...
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21 views

Metric evolving under Ricci flow with nonnegative scalar curvature is shrinking?

Let $g_{ij}(x,t)$ be a complete solution of the Ricci flow on a surface with bounded and nonnegative scalar curvature ,why the metric is shrinking ?
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26 views

What is $P\times_G E$?

I know what is principal bundle and associated bundle according Wiki.But I am not understand what is $P\times_G E$ .Seemly it is bundle,but I am not sure what structure is it . Below picture is from ...
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31 views

Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
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32 views

Why is this proof that a circular cone is not a surface not rigorous?

In example $4.1.5$, page $73$ of Pressley's Elementary Differential Geometry, a "heuristic" argument is given to prove that the circular cone with vertex the origin and angle $\pi/4$, is not a ...
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12 views

What is skew-product decomposition?

What is skew-product decomposition of Brownian motions referring to this paper Pauwels ,Rogers
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45 views

Equivariant flat U(1) bundle on a torus

I am trying to understand equivariant flat $U(1)$ bundles on a torus, $(S^1)^N$. By equivariant, I mean equivariance with respect to the natural action of $(U(1))^N$ on $(S^1)^N$. $G$-equivariant ...
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32 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
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31 views

Find the curvature of the metric $ds=\frac{|dz|}{(1+|z|^2)}$ on $\mathbb{C}$

The curvature of the metric $g$ is defined as $$k(z)=-\bigg(\frac{2}{\alpha(z)}\bigg)^2 \partial \bar\partial log \alpha(z)$$ where $\alpha$ is positive and real valued. Also ...
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36 views

An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that ...
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Confusion surrounding the Koszul-Malgrange theorem

On a recent project, I ran into something which seemed related to the Koszul-Malgrange theorem. According to nlab, the original statement of the theorem is Theorem 1. (Koszul-Malgrange theorem) ...
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Cokernal of the map $B_1 \times B_2 \rightarrow [B_1,B_2]$

I am reading Nakajima's book, Lectures on Hilbert Schemes of Points on Surfaces. In the proof of Theorem 1.9, it needs the following linear algebra fact. Suppose $B_1, B_2$ are two $n \times n$ ...
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30 views

Length of a differentiable curve with respect to a Riemannian metric.

Let $X$ be an $n$-dimensional differentiable manifold ($n\ge1$). A Riemannian metric in $X$ is a family $\{g_p\,|\,p\in X\}$, where for all $p\in X$: $g_p:T_pX\times T_pX\to\mathbb{R}$ is an inner ...
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49 views

Can you comb the hair on a 4-dimensional coconut?

It is well-known that you can't equip the surface of the unit sphere with a singularity-free coordinate system. Physicists have called this theorem (which is important for the theory of black holes) ...
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9 views

Sum two nearest function of two class are the nearest function of the sum class

Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ ...
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38 views

A question about Levi-Civita connection and curvature over 3 manifold

Give a 3-manifold M and Riemannian metric $g$, denote $A$ as the Levi-Civita connection on 3-manifold M corresponds to the metric $g$. Denote the curvature of $A$ as $F_A$, choose three bases ...
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19 views

Discrepancy between line integral over scalar field and line integral over vector field

There is a discrepancy between the line integral over a scalar field and the line integral over a vector field that is bothering me: Say $\gamma$ is a smooth curve. If $\gamma : \mathbb R\to \mathbb ...
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24 views

Vertical bundle of the pullback bundle

Let $\pi: E \rightarrow M$ be a smooth fibre bundle, $f \in C^{\infty}(N,M)$ for some smooth manifold $N$ and $f^{\ast}\pi: f^{\ast}E \rightarrow N$ the pullback bundle. How can I show the bundle ...
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27 views

Show that the tangent bundle is locally a product,i.e., $TU=U\times\mathbb{R}^{n}$.

Show that the tangent bundle is locally a product,i.e., $TU=U\times\mathbb{R}^{n}$. Where $TM$ is the set of pairs $(q,v),q\in M,v\in T_{q}M$, if $(U,x)$ is a coordinate system in $M$, then all ...
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22 views

Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
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39 views

Integral Curves of Gradient-like Vector Fields

If $X$ is a gradient-like vector field of a Morse function $f\colon M\to \mathbb{R}$, then the integral curve $c_p(t)$ starting at an arbitrary point $p$ approaches critical points as $t\to \pm ...
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Need help understanding this example of a distribution

Consider the following example of a distribution (given here): I tried to draw this. If $p=(a,b,c)$ then $$ X_p = (1,0,-b), Y_p = (0,1,0)$$ Then the planes in the distribution are planes spanned ...
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23 views

Integration over manifold

Let $M$ be a smooth 2-manifold in $\mathbb{R}^3$ such that $$4x^2+y^2+4z^2 = 4, y \ge 0$$ The boundary of $M$ is the set of points where $$x^2 + z^2 = 1, y = 0$$ Let $\alpha(u,v) = ...
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36 views

Terminology question about a weaker condition than normal crossings

Let $X$ be an algebraic set (say over $\mathbb C$). From what I understand, we say that $X$ has simple normal crossings if at every point it locally looks like a union of hyperplanes in general ...
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Is the restriction of a smooth vector field to a regular submanifold also smooth?

Let $S$ be a regular submanifold of a manifold $M$, meaning a subset of $M$ such that for all $p\in S$ there is a coordinate neighborhood $(U,\phi) = (U,x^{1},...,x^{n})$ of $p$ in the maximal atlas ...
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30 views

Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
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28 views

Multivariable Fundamental Theorem of Calculus

What's wrong with the following? Let $U$ be an open set in $\mathbb{R}^n$. Let $f$ be smooth on $U$. Define $$F(x_1,\cdots,x_n)=\int_{0}^{x_1}f(t,x_2,\cdots,x_n)dt$$ Then $f$ has an 'antiderivative' ...
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Cohomology of $\mathcal{O}^*$ and projection map

Suppose $X$ is a complex manifold and $T$ a complex space (or complex manifold maybe) and let $\pi:T\times X \rightarrow T$ denote the projection. What are sufficient conditions on $X$ that make ...
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Diffeomorphisms vs symplectomorphisms / volume conserving diffeomorphisms in an application

This question needs a bit of background: one way to study the mechanics of deformation of a continuous solid body is by defining a reference body $B_0$, a connected, well-behaved subset of $R^2$ or ...
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16 views

$\varphi$-related vector fields when $\varphi$ is an inclusion

I'm reading Jeffrey Lee's Manifolds and Differential Geometry. He's talking about vector fields being $\varphi$-related to each other. He says If $S$ is a submanifold of $M$ and $X \in ...
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28 views

Short question about differential forms / exterior algebra

I am working towards understanding the wedge product of two vectors. Here is what I have so far: Let $V$ be an $n$-dimensional vector space over $\mathbb R$ (or $\mathbb C$, I'm not sure it makes ...
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28 views

Tangent bundle of an almost complex manifold

Let $(X,J)$ be an almost complex manifold with dimension $2n$. Then the tangent bundle $TX$ can trivially be made an $n$ dimensional complex vector space along each fibre. But how can I find a smooth ...
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30 views

Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
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43 views

How to interpret $a_i^k g_{kj}g^{jl}$? (Tensor index notation)

I have the equation $$a_i^k g_{kj}=-b_{ij},$$ where $a_i^k$ are given by: $a_i^k \cdot\vec r_k=\vec n_i$. Here, $g_{ij}$ and $b_{ij}$ are the matrices that determine the first and second fundamental ...
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22 views

When does a Lie bracket exist for a Frechet manifold?

Does a general Frechet manifold admit a Lie bracket? A bracket can certainly be constructed in certain cases, but my guess is that it is wrong to assume that one exists in general.