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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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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28 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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40 views

Geodesic radius of curvature

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula. $\frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G}$ where $s$ is the ...
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46 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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28 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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16 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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22 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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16 views

Strainers in Alexandrov spaces

I am reading the section on Strainers in Burago, Burago and Ivanov's book "A Course in Metric Geometry". I have been struggling with the proofs of some of the lemmas. On Lemma 10. 8. 13, the authors ...
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56 views

Accepted symbol (or way of writing) “A is a subset of B or B is a subset of A”

I am looking for a concise way to write the statement "$A$ is a subset of $B$ or $B$ is a subset of $A$". The context is the Grassmannian and two elements $A,B\in G_k(\mathbb R^n)$ in it. The two ...
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24 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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31 views

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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23 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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29 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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29 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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36 views

Riemann Sphere: Holomorphic Functional Calculus

Why do we consider the holomorphic functional calculus on the Riemann sphere rather than the complex plane only? Is there a serious problem? Moreover isn't any curve encircling the spectrum ones ...
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45 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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34 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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22 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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18 views

If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler > 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 Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
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11 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
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11 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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23 views

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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32 views

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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41 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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24 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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50 views

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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19 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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36 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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54 views

Stokes theorem on manifold without boundary

I'm struggling to understand why the integral should vanish: $\partial M=\varnothing:\quad\int_Md\omega$ For example: $0=\int_{B_1(0)}d(ydx)=\int_{B_1(0)}1dy\wedge dx=\mu(B_1(0))\neq 0\text{ ?}$
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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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55 views

Given a closed form, show it is exact

I came across an old exam problem and I wonder how to approach it: 1) Show that if $f$ is a $k$-form on $\mathbb{R}^{n}$ and $df=0$, $k>0$ then $f=dg$ for some $k-1$-form $g$. My first thoughts: ...
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20 views

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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21 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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51 views

Flow of Linear Vector Fields

The following is a statement from "Notes on the Topology of Vector Fields and Flows" by Daniel Asimov. In the case where a vector field on $\mathbb R^n$ is defined by a matrix, then there is a simple ...
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21 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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30 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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19 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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37 views

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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33 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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40 views

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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38 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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27 views

References for Conjugate Points in Differential Geometry

I will have to give some lectures about conjugate points and I need some nice references about it, can anyone recommend me some? I already know manfredo's differential geometry of curves and surfaces ...
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45 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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17 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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131 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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19 views

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.
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29 views

dimension of tangent space to a boundary point of a convex shape

I have a basic question regarding the dimension of the tangent space at a point $P\neq0$ that lies on the boundary of a pointed convex cone with its point centered at 0. For a 3D cone that is ...
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26 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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39 views

Notation on an estimate of the sectional curvature.

In a paper on the Ricci flow i am currently reading (http://arxiv.org/abs/math/0612095) the following estimate occurs several times (for example Lemma 4.1 and 4.2); $$\operatorname{sec}(g_0) \geq ...
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93 views

How do i show that dy/dx does not equals to zero

The question goes like this. Equation of a curve is $2x^2-3xy+y^2=5$ Find the equations of the tangent and normal to the curve at point $(4,3)$. Show that there is no point on the curve at which the ...