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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Equivalent descriptions of “flat space with non Euclidean metric” and “curved space with (local) Euclidean metric”: the case of Minkowski space.

FIRST: I start with the guiding idea: 1. we have the surface of a paraboloid (z = x2 + y2); its metric, in an infinitesimal neighbourhood of one of its points is (we can choose it) EUCLIDEAN; now, ...
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Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
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32 views

Connection defined by its geodesics

I have a question related to the definition of a connection (in the sense of Koszul) by its geodesics. I know that a torsion free connection is uniquely determined by its geodesics. Now, let $M$ be a ...
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41 views

Coordinate frames along the bounday of a minimal area (soap-film) surface

I would like to calculate coordinate frames along a closed Bezier (Or Catmull-Rom) spline. One axis should be tangential to the curve, and another axis normal to the minimal-area surface (soap-film ...
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86 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
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55 views

Derivative with respect to a vector and tensor on a manifold

I am reading through a paper and have come across a statement which I do not fully understand. I paraphrase below. Consider a scalar function $f = ...
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21 views

A construction on principal bundles

In a paper the principal $Sp(1)$-bundle $P$ over $S^4$ is introduced as follows: let $Sp(1)\times Sp(1)\hookrightarrow Sp(2) \xrightarrow{\pi} S^4$ be the spin structure on $S^4 $. The principal ...
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71 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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19 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
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24 views

curvature of a curve varying with time

Just wanted to know is there any literature available for the following problem. I have two points in 2d and I want an actuator to pass through both points. I want to find how the curvature varies ...
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35 views

curvature and torsion information from the 3d points

Hi I have some n points in 3d space and I want to connect them to form a 3d curve. I want to know a method of finding its torsion and curvature along its arclength. Any references or example would ...
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31 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
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34 views

How does the reduction of the frame bundle affect the tangent bundle

Let $M$ be a differential manifold and $F(M)$ its frame bundle. Suppose there is a reduction of the structure group of $F(M)$ from $GL(m,\mathbb{R})$ to the Lie group $H$ and let $F_{r}(M)$ be the ...
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33 views

Reference frame rotation depending on metric tensor $g_{\mu\nu}$

My simple question is: The infinitesimal line element is $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$$ where $g_{\mu\nu}$ is the metric tensor of the space. Is it possible from the simple knowledge of ...
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31 views

Lower boundary for hessian eigen values

This question appears on a differential geometry test i am trying to solve. Let $f$ be a function of $x,y$ and $k>0$ a positive real number. the function obeys the following: $f(0,0)=0$ and ...
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50 views

What is $T^*(\mathbb{A}^1)$?

Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
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31 views

Does a metric has a neighboorhood in which signature is the same?

suppose $M$ is a manifold and $g$ is a metric with a specific signature on it. there are at least 2 topologies for metrics that I am aware of. Fine $C^k$ topologies and the topology of metrics as ...
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41 views

Geodesics on a cone satisfying a certain condition.

Consider the curve $\gamma$ with curvature given by $$\kappa = \dfrac{a}{s^2 +a^2}$$ and torsion given by $$\tau = \dfrac{s}{s^2+a^2},$$ where $a>0$ is a constant.\ It turns out that the ...
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25 views

Geometric significance of a certain dot product on a ruled surface.

On a non-developable ruled surface generated by the principal normals to a curve, the striction curve intersects the normal at the central point. If $P,\,Q$ are any pair of points on a normal such ...
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32 views

Terminologies for induced connections

Given a Riemann manifold with a Kozul/Affine connection, if you take any subbundle of the tangent bundle there is an induced connection given by applying the ambient Kozul connection and projecting to ...
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Dimensions of the cohomology groups of certain complicated space

Let contruct the space $X$. We take the complex projective space $\mathbb{C}P^2$, pick two points $p_1, p_2 \in \mathbb{C}P^2$ and remove two small, disjoint, open $4$-balls $B_j$ centered at $p_j$. ...
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29 views

Differentiable structure on the gauge group?

In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369). Let $P\rightarrow M$ be ...
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37 views

Find the areas $A(D)$ of $D$ and $A(\sigma(D))$ of $\sigma(D)$.

A regular parametrized surface $\sigma:U\to\mathbb{R}^{3}$ is given where $U=\lbrace (x,y)\in \mathbb{R}^{2}\mid (x,y)\neq (0,0)\rbrace$. Only the coefficients of the first fundamental form are known ...
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40 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
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75 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
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47 views

Why in a local holomorphic trivialization , connection compatible with the holomorphic structure have the form $\nabla ^A=d+A$

Let $X$ be a complex manifold and $L\to X$ a holomorphic line bundle. Why in a local holomorphic trivialization , connection compatible with the holomorphic structure have the form $$\nabla ^A=d+A$$ ...
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26 views

Extension of a bounded operator on manifold

I have a problem, which is quite urgent. I am given a pseudodifferential operator $A$ in the space $L^0_{\rho,\delta}(M)$, where $M$ is a compact manifold. I wish to extend this operator to an ...
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21 views

If $M$ has hyper-Kähler structure then $M//G$ has hyper-Kähler structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kähler manifold, then the symplectic quotient of $M$, i.e, ...
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16 views

existance of lorentzian metric sequence

Suppose M is a manifold and $g, h$ are lorentzina metrics on it. We write $g < h$ iff the null cone of $g$, including its null vectors is a subset of the null cone of $h$ at every point $p$. i.e. ...
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the induced verctor field of the compose map

Let $M$ be a manifold, $\phi_t, \varphi_t: M\to M$, they induce two vector field, $$ \frac{d}{dt}\phi_t=X_t\circ \phi, \frac{d}{dt}\varphi_t=X_t\circ \varphi, $$ then what's the vector field induced ...
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Why is this flow (no) complete

I'm having trouble understanding complete flows. The deffinition of a flow I'm using is that a flow of a vector field defined in a manifold is complete if and only if for any point of the manifold the ...
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80 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
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45 views

There are no conjugate points on a surface with negative Gaussian curvature?

I'm trying to understand the following theorem about conjugate points: Theorem. Let $M$ be a complete surface with Gaussian curvature $K\leq 0$, then there are no conjugate points on $M$. Proof: Let ...
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Calculate the geodesic of Z = XY between two points

I have only learned about calculus and linear algebra, so I don't know about differential algebra. I got to know about the concept of "geodesic" recently. What I need to know is this: Suppose I ...
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39 views

How many inflection points can a spatial parametric cubic curve have?

So in my knowledge, a planar cubic polynomial curve always has one and only one inflection point. My question is, for a spatial parametric cubic curve parameterised by t, which has the equation of: ...
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44 views

Ricci curvature tensor calculation?

Is this correct? $$ R=R_{ab}g^{ab} $$ $$ R(g_{cd})=R_{ab}g^{ab}(g_{cd}) $$ $$Rg_{cd}=R_{ab}g^{ab}(g_{ab}\delta^a_c\delta^b_d)$$ $$Rg_{cd}=R_{ab}(g^{ab}g_{ab})\delta^a_c\delta^b_d$$ ...
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Determining an arclength and a unit vector - asking for corrections.

I'd like you to help me answer/solve the two problems. There are also my attempts. I'd be appreciated if you could correct them and help me doing the right way. Let $\sigma:U\to\mathbb{R}^{3}$ be a ...
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derivative of the composition of two smooth map is the composition of derivatives

I can't type the LATEX so I uploaded the image. By Googling, I found that is the result of the pushforward, but there is no proof so I can't understand. ...
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30 views

How could I calculate displacement along a 2D polyline by integrating each dimension separately?

This question's field of application is GPS trajectory analysis, but I'll try to give it a more abstract mathematical treatment. Suppose the trajectory of an object in 2D plane is described by a ...
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32 views

A question on self-adjointness of the shape operator

In the book Elementary Differential Geometry of Christian Bar, CUP, on the page 108, of the proof of theorem 3.5.5, the author wrote: Theorem: Let $S ⊂ \mathbb{R}^3$ be an orientable regular surface ...
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60 views

Understanding QEF's and assorted techniques for dual marching cubes

I'm attempting to implement dual marching cubes in a 3d engine and I'm getting lost in the math portion. http://www.cs.rice.edu/~jwarren/papers/dmc.pdf is the link to the original white paper. Part ...
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24 views

Local coordinates for a hypersurface

Let $M$ be a hypersurface in Euclidean space $E^{n+1}$, $e_1,e_2,..., e_n$ a frame field for its tangent bundle and $\theta_1,\theta_2,...,\theta_n$ are dual base. By knowing that $\theta_1$ is ...
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Theorema Egregium

I'm having trouble understanding what the Theorema Egreguim is. I read stuff on it but I'm not really grasping it. I know that it ties in with the Gaussian curvature but is there like a specific ...
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38 views

Meaning of fibered product

I need a small explanation about the next. If we write $p: TM\to M$ for the natural projection and $F$ for the natural bundle with $FM=p^{*}(T^{*}\otimes T^{*})M\to M$, then ...
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Foliation preserving diffeomorphisms for a codimension 1 foliation

I am studying reference frames on Minkowski spacetime $\mathcal{M}$, with (+,-,-,-) signature, from a differential geometric point of view, for this reason I came up with (codimension 1) foliations ...
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39 views

Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold? Explanation Suppose we have a 3D smooth manifold ...
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94 views

Strictly convex boundary of Riemannian manifold

Let $(M,g)$ be a compact smooth Riemannian manifold with boundary $\partial M\subset M$. What does it mean to say that the boundary is convex and strictly convex? I can find definitions of ...
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23 views

spin^c structures and charged spinors

Given a spin structure and a complex line $\mathcal{L}$ we can form the tensor product of the complex spinor bundle $S$ and this line $S\otimes\mathcal{L}$. A spin^c structure attempts to construct ...
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136 views

Periodicity of Ideal Packings on Arbitrary Boundaries

This is a question that's been nagging me for some time that I find particularly interesting, but have no idea how to go about solving it. Consider some arbitrary solid $S$, such that $\partial S$ is ...
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Isotropy of transitive lie algebroid

Let $\rho:E{\longrightarrow} TM$ is a transitive lie Algebroid. I wanna show $Ker\rho$ is a vector sub-bundle of $E$ by introducing its bundle-chart. please hint me.