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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Does positive definite Hessian imply the Jacobian is injective?

Suppose $f(x):\mathbb{R}^n \mapsto \mathbb{R}$ is an infinitely differentiable function. If $\nabla^2 f(x)$, the hessian of $f$ is positive definite everywhere, does this imply that the gradient(first ...
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Symbol of the differential operator on vector bundles

Suppose that we have a differential operator $D:C^{\infty}(\mathbb{R}^n) \to C^{\infty}(\mathbb{R}^n)$ of the form $(Df)(x)=\sum_{|\alpha| \leq k}a_{\alpha}(x)\frac{\partial^{|\alpha|}f}{\partial ...
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Solution manual for Modern Differential Geometry for Physicist? [closed]

Here is the book by Chris J. Isham Anyone has the solution manual of Modern Differential Geometry for Physicist?
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Please could someone check and help me with my answer to part two of this exercise about vector fields along maps?

I previously solved the following (first half of an) exercise: Let $U$ be an open neighbourhood of $0 \in \mathbb R^2$ and let $f = (f_1, f_2) : U \to \mathbb R$ be a smooth map such that $f(0) = ...
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19 views

Please would someone check my answer to this exercise on vector fields along maps?

I believe I solved the following exercise and would appreicate it greatly if someone could check my answer: Let $U$ be an open neighbourhood of $0 \in \mathbb R^2$ and let $f = (f_1, f_2) : U \to ...
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28 views

The literature on Chern-Simons theory

Can any one give some literature on Chern-Simsons theory? I can not find any book introducing this theory. Thanks.
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32 views

'Unrolling' the neighbourhood of a space curve

I have a space curve $\gamma : \mathbb{R} \longrightarrow \mathbb{R}^3$, sampled at $n$ discrete points. I have implemented an algorithm that gives me an approximation to $\gamma$'s tangent, normal ...
3
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35 views

Smooth maps preserve dimension

I stumbled over a useful consequence, that is apparently wrong for only continuous maps. Imagine $A \subset \mathbb{R}^{n-1}$ is a compact set and $F : \mathbb{R}^{n-1} \rightarrow S^{n}$ a smooth ...
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64 views

Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
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3answers
113 views

On defining homology groups

I have been trying to understand what homology groups are "talking about," and now I am wondering if the following works as a definition of homology. But first, some illustration of what it is ...
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1answer
37 views

Is the inside of a sphere a hyperbolical surface?

So since an elliptic surface with constant curvature would be a sphere, would a an hyperbolical surface with a constant curvature be the inside of a sphere if we were to go out from inside the sphere? ...
3
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33 views

Level set as the orbit of the action of a Lie Group?

I'm wondering the following. Given a smooth function $f:\mathbb R^n\rightarrow \mathbb R^m$ with $m<n$ and level sets $\mathcal O(y)=\{x\in\mathbb R^n| f(x)=y \}$. What are the conditions on $f$ ...
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+50

Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \begin{equation} \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...
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17 views

geodesic of Stiefel manifold

Define a metric on Stiefel manifold $V_{n,p}$ as $$\left<\Delta_1,\Delta_2\right>=\text{tr}\Delta_1^T\left(I-\frac{1}{2}YY^T\right)\Delta_2$$ $\forall \Delta_1,\Delta_2\in T_YV_{n,p}$ how to ...
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1answer
10 views

What are intuitively the diffeomorphisms $\theta^t(p)=\Theta (t,p)$, associated to the local flux $\Theta$ of the vector field $X$?

Given a vector field $X$ on the manifold $M$ I know that I can associate to it in a unique way a local flux $\Theta: W \rightarrow M$, where $W \subset\mathbb{R} \times M$. The curve ...
3
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1answer
60 views

In manifold theory, in what sense is the derivative a first-order approximation?

As I move on from the calculus definition of the derivative to the differential geometric definition in terms of tangent spaces, I am wondering how to recover the notion that the derivative of a ...
2
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2answers
64 views

Vector fields along maps: I need another sanity check

Consider the definition of a vector field along a smooth map $f: M \to N$ where $M,N$ are smooth manifolds: A vector field along $f$ is a continuous map $W \colon M \to TN$ such that $W(m) \in ...
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1answer
19 views

Symplectic manifold

Let ($M$, $\omega$) be a symplectic manifold of dimension $2n$. Then $\omega$ is non-degenerate $2-form$ by definition. Now, my question is if we can conclude that $\omega \wedge ...\wedge \omega$ ...
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Defining an Ehresmann Connection Using Linear Connection

In some places that I have seen, given the covariant derivative $\nabla$ of a linear connection on a vector bundle $\pi: E\rightarrow M$ (dim(M)=k), an Ehresmann connection is defined in the following ...
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19 views

Why $exp(0_{T_eG})=e$, where $exp$ is the exponential map of a Lie group?

I wonder if this fact is true: I consider the exponential map of a Lie group $G$. $$exp: \mathfrak{g} \rightarrow G.$$ Is it true that $exp(0_{T_eG})=e$, where $e$ is the identity element of $G$? ...
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47 views

Question on vector fields along maps (need a quick sanity check)

Let $f: M \to N$ be some smooth map between smooth manifolds. If $V$ is a vector field, that is, a smooth map $V: N \to TN$ then $V$ is a vector field along $f$ if the projection $\pi: TN \to N$ is ...
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1answer
38 views

Differential structure on the cone

Let's take $\mathbb{R}^2$ with the action of a cyclic group by standard rotations with center the origin. Can we put on the quotient a differential structure such that the projection is ...
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1answer
38 views

Basic examples topological manifolds with boundary

I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...
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36 views

Explicit Calabi-Yau metrics

I would like to know which explicit metrics on noncompact Calabi-Yau (CY) threefolds are known. For instance, an important class of such spaces can be constructed algebraically, including local ...
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1answer
24 views

Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point. The background So he wants to show that any symplectic form is ...
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35 views

Topology of statistical manifolds

I am currently working with statistical manifolds. Roughly, a statistical manifold is a set of distribution parametrized by a set of parameters. However i have trouble finding more precise definition. ...
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3answers
46 views

Group actions on manifolds - exponential map

Let $M$ be a smooth manifold. Suppose $K$ is a Lie group (with Lie algebra $\mathfrak{k}$) acting EDIT: TRANSITIVELY on $M$ from the left and $G$ is a Lie group (with Lie algebra $\mathfrak{g}$) ...
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11 views

Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
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290 views

Example of a surface where more than one coordinate patch is needed.

I find the sphere example underwhelming. Sure I can see that one open patch will not cover it, but it still manages to cover it mostly. So much so that you can go ahead and, say, calculate the area of ...
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3answers
69 views

Exterior derivative

Let $$\omega = \frac{1}{2} \sum_{i,j} \omega_{i,j} dx_i \wedge dx_j$$ be an antisymmetric form, so i.e. $\omega_{i,j} = - \omega_{j,i}.$ Now, in some lecture notes I found that $$d \omega ...
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1answer
115 views

What is the equation describing a three dimensional, 14 point Star?

I need to model a 14 point star. This is a three dimensional surface where there is a point at each of the eight corners of a cube and each of the six sides. The object is uniform (i.e. planar ...
4
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108 views

Fiber bundles with category morphisms as fibers

Given a total space $E = M \times F$ of a fiber bundle where $M$ is a smooth manifold and $F$ is the fiber. The fiber $F_x$ corresponding to the point $x \in M$ is the set of morphisms between objects ...
2
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2answers
39 views

Dimension of the $0^{\text{th}}$ de Rham cohomology group of $U$

Let $U\subset\mathbb{R}^n$ be an open set. I proved that $$ H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \} $$ I have to show ...
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$g_{ij}$ calculation of randers metric

Let $F=\alpha + \beta $ where $\alpha=\sqrt{a_{ij}(x)y^iy^j}$ is a riemannian metric and $\beta =b_i(x)y^i$ is a one form.that is F is Randers metric on a manifold $M$. I want to calculate $g_{ij}$ ...
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1answer
14 views

Critical and regular values of height functions on a closed hypersurface

Let $M$ be a closed connected hypersurface of $n$-dimensional in $\mathbb{R}^{n+1}=\{(x^1,\cdots,x^{n+1})\}$ and let $\nu$ be a smooth unit normal vector field of $M$ at $\mathbb{R}^{n+1}$, $H$ be the ...
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38 views

geometry of the sphere

I wish to understand the geometry of the sphere so that I can work on it for PDE problem. Could anyone suggest some good references for this (notes/books etc)? thanks
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1answer
37 views

Symplectic geometry spectrum

I have a question about a step in the proof of the following theorem from symplectic geometry: The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le ...
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1answer
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Tangent space manifold

Let M be a differentiable manifold of dimension m and also let $\{\xi_1,\dots,\xi_m\}\subset \text{T}_pM$ be an linearly independent set of the tangent bundle of M at a certain point $p\in M$. I have ...
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1answer
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Determine the exponential map of the direct product of two Lie groups.

I know that the direct product of two Lie groups $G$ and $H$ is a Lie group. Knowing the exponential map of $G$ and $H$, I would like to find an expression for the exponential map of $G \times H$. ...
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28 views

Equivalence of two integral conditions

Consider the following parametrization of the unit ball in $\Bbb{R}^3$: \begin{align}T:(0,1)\times (0,\pi)\times (-\pi ,\pi)&\to \Bbb{R}^3 \\ r,\theta,\phi &\mapsto(r\sin \theta\sin \phi, r ...
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2answers
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Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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How to tell whether a curve has a regular parametrization?

A parametrization of a 1-dimensional curve is called regular if its velocity is always positive. For example, the following parametrization: $$x(t)=t^3, y(t)=t^6$$ is not regular because its ...
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Is there any reason why both derivatives should be non zero?

Let $f = (f_1, f_2) : \mathbb R, 0 \to \mathbb R^2 , 0$ be smooth and such that $n = \min (ord(f_1), ord(f_2)) < \infty$ where $ord(f) = \min \{n \in \mathbb N_{>0}\mid {\partial^n \over ...
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Curve shortening flow and strong maximum principle

I am in particular uncertain about how the strong maximum principle is used in the argument below. Could someone please clarify and add more detailed explanations. Thanks So assume we have a regular ...
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1answer
24 views

Hamiltonian vector field and symplectic geometry

I want to show the following theorem: For any Hamilton function $H : M \rightarrow \mathbb{R}$ on some symplectic manifold $M$ and symplectomorphism $f : M \rightarrow M$ we have $X_{H \circ f} = ...
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2answers
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Euclidean metric on a Riemannian manifold

Lets say we have a Euclidean configurations space $\mathbb E^n$ equipped with a smooth inner product $\langle \cdot ,\cdot \rangle$ with positive signature in the tangent space above each point. We ...
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1answer
41 views

Are there noncontinuous derivations $C^1(X) → ℝ$?

I’m looking for an example of a Banach space $X$ and a derivation $δ \colon C^1(X) → ℝ$ which is noncontinuous with respect to the topology of uniform convergence on $C^1(X)$, that is a $ℝ$-linear map ...
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1answer
37 views

Is pointwise multiplication by a smooth non zero function a diffeomorphism

Say $f: \mathbb R \to \mathbb R$ is nowhere zero (like e.g. the constant map 1). Is the map $x \mapsto x f(x)$ a diffeomorphism? It seems to me that the answer is no because the derivative of a ...
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1answer
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Lie bracket is part of the intrinsic “geometry”? But I have seen it defined without a metric…?

I have seen two definitions of the Lie Bracket for a Riemannian manifold $(M,g)$. One is this : $[X,Y] = D_X Y - D_Y X$, where $D$ stands for covariant differentiation. When written out, this seems ...
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1answer
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Transformation behavior of connection on vector bundle.

Using the notation from Jost's various books on geometry, let $$ D=d+A $$ be a connection on a vector bundle $\pi:E\rightarrow M$ with structure group $GL(n,\mathbb{R})$. Also let $\{U_\alpha\}$ be ...