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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When is the metric completion of a Riemannian manifold a manifold with boundary?

Let $(M,g)$ be a connected smooth Riemannian manifold and denote by $(M,d)$ the induced metric space following by taking topological metric to be the infimum over length of curves in the standard way. ...
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31 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
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1answer
48 views

For which Lie groups $G$ can one write $g$ as the exponential $\exp X$ of some $X \in {\frak g}$ for every element $g \in G$?

I am reading a book on matrix Lie algebras (Brian Hall's). Corollary 2.30. says that if $G$ is a connected matrix Lie group, then every element $A$ of $G$ can be written in the form ...
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19 views

geometric meaning of conjugate points

Recently I am reading Manfredo do Carmo's Differential Geometry of Curves and Surfaces. He said the $q$ is the conjugate point of $p$ with respect to a geodesic $\gamma$ joining the two points if ...
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1answer
34 views

Proof that the special linear group $\mathrm{SL}(n,\mathbb{R})$ is a smooth manifold

This is my definition of a smooth manifold I am supposed to work with: Let $\mathcal{M} \subseteq \mathbb{R}^{n}$. The set $\mathcal{M}$ is a $k$-dimensional smooth submanifold of $\mathbb{R}^{n}$ ...
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1answer
49 views

Equivariant generalization of $\mathcal{O}(1)$ on $\mathbb{CP}^1$

The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by \begin{equation} A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2} \end{equation} This follows since ...
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+50

Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
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evaluate this region using gauss's theorem (only using the triple integral 'part')

Evaluate $$\iiint _{D}\vec{\nabla} \cdot\vec{F}\,dV$$ with $$\vec{F}=\left \langle x^{2},y,z \right \rangle$$ $$D=\left \{ \left ( x,y,z \right )|x^{2}+y^{2}+1\leq z\leq 5 \right \}$$ ...
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23 views

Proving $\mathrm{GL}(n,\mathbb{R})$ is a smooth manifold

Consider the set $\mathrm{GL}(n,\mathbb{R}) = \{ \ A \in M_{n \times n}(\mathbb{R}) \ | \ \mathrm{det}(A) \neq 0 \ \}$. I'm trying to show that this is smooth submanifold of $\mathbb{R}^{n^{2}} \cong ...
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1answer
56 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
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1answer
33 views

Explicit procedure for integrating densities (twisted $n$-forms)?

I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. ...
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26 views

Parametrisation of the curve after a short time

I am trying to wrap my head around this differential geometry problem. I am given velocity V with components in the principle normal and binormal directions. Then I am given an approximation of the ...
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20 views

Upper bound on hessian

Given a smooth Riemannian manifold $(\mathcal{M},g)$ and $f \in C^{\infty}(\mathcal{M})$ let $r(x)= d(x,x_0)$ where $d$ is the distance function wrt $g$ and $x_0$ is some point on the manifold. If we ...
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1answer
42 views

Symmetric Curve

I am having hard time solving the following question: Let $\gamma : \mathbb{R} \to \mathbb{R}^2$ curve in natural parameterization. Suppose the curve has curvature $\kappa(s)=3s^2$ Then ...
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1answer
28 views

Darboux coordinates on projective spaces

I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of ...
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1answer
33 views

How to denote raising $x^1$ to a power in differential geometry

I'm working from a text in which the coordinates of a point in $\mathbb R^n$ are denoted $(x^1,\dots,x^n)$. I'm wondering if there is a standard way to denote the sum of the squares of these ...
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1answer
29 views

How to prove this orthogonality? [duplicate]

In $\mathbb{R} ^3$, $v$ is a fixed vector and $\alpha:I\rightarrow \mathbb{R}$ is a regular curve. We have: $\alpha'(t)$ is orthogonal to $v$ for all $t\in I$ and $\alpha (t_0)$ is orthogonal to $v$. ...
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30 views

Does the forgetful functor from smooth manifolds to sets preserve colimits?

It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits? It certainly preserves all examples of ...
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29 views

Diameter of closed curve is perpendicular to velocity?

Let $\gamma$ be a curve in $\mathbb{R}^2$ and $d:\mathbb{R^2}\times\mathbb{R}^2\to\mathbb{R}$ be the distance function between two points of $\gamma$. I'm trying to show that if the distance between ...
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22 views

lower bound on volume of balls

It is well known that a lower bound on Ricci curvature gives an upper bound on the volume of balls. What are conditions that gives a lower bound on the volume of balls? It is reasonable to think that ...
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1answer
13 views

Help with understanding a proof of compact surface having an elliptic point

In my studies of differential geometry from do Carmo's book, I have come across a very nice claim which states that a regular compact surface has an elliptic point that is a point with positive ...
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1answer
130 views

Why is $s$ used for arc length?

Why is $s$ used for arc length? I looked around online, but I can't find a definite answer. Thank you!
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27 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
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24 views

Value preservation along geodesics.

Given a differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$, where $n>m$. We define the graph of $f$ $ W_f = \{ (x,y) | x \in \mathbb{R}^n, y \in \mathbb{R}^m,y=f(x) \}. $ Given two ...
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21 views

Homotopic curves

I'am studying differential forms and I've arrived at the theorem of invariancy of integrals of closed forms over homotopically equivalent curves. I came up with a problem: If I take a curve with ...
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23 views

Prove: A unit speed curve $\alpha(s)$ with $\kappa>0$ is rectifying if and only if $\frac{\tau}{\kappa}=as+b$

Given a curve $\alpha$, the plane determined by {$T,B$} is called the rectifying plane. A unit speed curve is said to be rectifying if for all $s$ and for some constant $p$, $\alpha(s)-p$ lies in the ...
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51 views

An alternative derivation of radius of curvature (2D functions). How valid is it?

I was wondering how radius of curvature was derived, and this is what I came up with. It turned out to be longer than expected. Then I looked at how it compares with other (presumably more ...
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1answer
27 views

Showing 0 curvature

I'm trying to show that a curve, $\gamma$, of general type in $\mathbb{R}^n$ that is contained in an $(n-1)$-dimensional affine subspace has curvature $\kappa_{n-1}=0$. My thought process was to ...
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1answer
35 views

why is $\dfrac{dr}{r~d\theta} = \cot \psi$?

why is $\dfrac{dr}{r~d\theta} = \cot \psi$ ? Extracted from Ordinary Differential Equations, Garrett Birkhoff, in the chapter of Linear Fractional Equations (First order Differential Equations). ...
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31 views

How does one determine whether a coordinate basis is orthogonal or not?

Apologies for what is perhaps a very basic question, but I have been studying differential geometry with a view to gain a deeper understanding of general relativity and I have hit a stumbling block. ...
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32 views

Is any smooth deformation of a metric in dimension 1 conformal?

Consider $(S^1, g)$ where $S^1$ is the unit circle and g is a metric. Now consider the metric $$ \tilde g := f g $$ where f is a smooth positive function. Since in 1 dimension this is the only smooth ...
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Regarding the axis of screw motion for a space curve.

The axis of the accompanying screw motion of a curve $c(s)$ at any point $c(s_0)$ is the line in the direction of the Darboux vector $\tau(s_0) T(s_0) + \kappa(s_0)B(s_0),$ through the point $$P(s_0) ...
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2answers
44 views

What are some interesting uses for/motivations of projective spaces?

I have trouble motivating myself to think about real projective spaces, for instance. Are there any cool results about them? Are there any motivating examples?
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35 views

Rank of Jacobian Matrix for the Stereographic Projection

With the definition $S^{n} = \{\ \mathbf{x} \in \mathbb{R}^{n+1}\ | \ ||\mathbf{x}|| = 1\ \}$, and the function $\ f:\mathbb{R}^{n} \to S^{n} \setminus \{ (0,...,0,1) \}$ defined by: $f(\mathbf{u}) ...
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1answer
36 views

Is the associated bundle construction a bifunctor?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and ...
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Cohomology algebra generated by Steifel-Whitney classes and dual classes subject to defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
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24 views

Hermitian Structure on $\mathcal{O}(1)$ line bundle over $\mathbb{P}^{n}$

I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can ...
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Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that ...
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2answers
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Geodesics on surfaces of revolution about z axis with negative curvature

This is a question in differential geometry of surfaces that I could not do We are given S a surface of revolution about the z axis with everywhere negative Gaussian curvature. We are to show that the ...
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27 views

Why is the backwards curve needed for the inverse of a parallel transport?

Dearh math.stackeschange-community, I'm at a loss with the following problem: Let $I=(a,b)$, then the inverse of the parallel transport $P_{s,t}^\gamma$ from $s \in I$ to $t \in I$ along the curve ...
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1answer
72 views

Lie groups pre-requisites and reference

What are the minimum pre-requisites in analysis (differential geometry) required to study Lie-groups? And for that material, what are some good references? I have done basic courses in Metric spaces, ...
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1answer
14 views

Geodesics on surface of revolution of regular curve

I was recently presented with this in differential geometry stating the following: Let us define the regular curve on the XZ plane as: $ \gamma (t) = (sin(t)+2,0,t) $ on XZ plane for $ t \in R $, ...
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Complex Hopf Fibration

The Hopf construction gives a circle bundle $p$ : $S^{3}$ → $\mathbb{CP}^1$. The equation of a 3-sphere in $\mathbb{R}^4$ is $X^2+Y^2+V^2+W^2=R^2$, where $R$ is the radius of the 3-sphere. We may ...
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Showing that $(\mathbb{R}, \mathscr{F})$ and $(\mathbb{R},\mathscr{F_1})$ are diffeomorphic but $\mathscr{F}\neq \mathscr{F_1}$

Background $M$ is locally Euclidean with dimension $d$ if $M$ is hausdorff and every point in $M$ has a neighborhood homeomorphic to $\mathbb{R{^d}}$. If $U\subset M$ is open and connected and $\phi$ ...
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39 views

Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
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28 views

A step in proof of Bishop-Gromov Theorem

I am reading the proof of Bishop-Gromov's comparison theorem in Schoen and Yau's Differential Geometry book. They do it using the Jacobi fields approach. There is a step which I have trouble ...
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19 views

Number of Frenet Frames

How can I show that a regular curve in $\mathbb{R}^n$ has $2^n$ Frenet frames? I guess intuitively I'm thinking that since a Frenet frame is a moving reference of $n$ orthonormal vectors and each ...
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1answer
22 views

Are the symplectic leaves of a Poisson manifold submanifolds?

In "Introduction to Mechanics and Symmetry" by Marsden and Raţiu it is written, in chapter 10, page 347, example b, that "[s]ymplectic leaves need not be submanifolds". In "Lectures on Poisson ...
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29 views

derivative of a projection matrix

The projection onto a parametrised vector $v(\lambda)$ is $P_v = \frac{vv^{T}}{v^{T}v}.$ Its complement is $$P = I-\frac{vv^T}{v^{T}v}.$$ I've got an expression containing this complementary ...
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1answer
24 views

Christoffel symbols of $S^n$ in polar coordinates

Consider the usual local polar coordinates $\theta_1, \theta_2,..., \theta_n$ on $S^n$. We were taught about Christoffel symbols today and I am trying to see what the Christoffel symbols of $S^n$ ...