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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Orientability and volume form

I was wondering if there is an easy argument that manifolds that are orientable always have a volume form and vice versa? So I am not looking for a full proof of this, but rather a good argument how ...
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78 views

gluing manifolds along boundaries

I have a problem with the following task. Suppose $M_1, M_2$ are smooth manifolds with boundary. Let $f_1, f_2$ be diffeomorphisms from $B_1$ to $B_2$ ($B_i$ - the boundary of $M_i$), and suppose ...
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60 views

Is there a generalization of the Quaternionic Hopf fibrations and its natural connection?

For the quaternionic Hopf fibrations, $S^3\rightarrow S^7\rightarrow S^4$, we have a natural BPST connection form. Do we have some generalization of it? For example, the 'natural' connection form on ...
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78 views

an isomorphism between the tangent space of a manifold to euclidean space

1) I was told in class many years ago that, the tangent space if the sphere $\mathbb{S}^2$at a point $p$, i.e. $T_p\mathbb{S}^2$ is isomorphic to $\mathbb{R}^2$. Could anyone give me a proof of ...
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69 views

About the diameter of a Riemannian manifold

My exact problem is a Riemannian manifold defined on $SU(2^n)$, where the metric is defined as follows: If $U(t)$ is a curve so that $U′(t)=−iH(t)U(t)$ (so just a unitary evolution of a quantum system ...
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55 views

Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

I'm looking for an introductory book in differential topology in which there are proofs of the following facts: Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ ...
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46 views

Integral curves on non compact manifolds

Define a vector field on $\mathbb{R}^d$ by $X = \frac{\partial}{\partial x_{d}}$. That is a vector field that always points upward along the $x_{d}$-axis. Consider starting at any point $p \in ...
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44 views

Preserving arc length for a family of curves

Sorry if I have formatted things wrong, I have read the tour and browsed around, so I tried my best. I have a one parameter family of curves with the relation: $$\frac{\partial}{\partial \lambda} ...
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41 views

Show that $1$-form has particular coordinate representation.

I want to show that for a nowhere-vanishing $1$-form $\alpha$ on an $n$-dimensional manifold $M$ we have that $\alpha \wedge d\alpha = 0$ if and only if around every point $p$ there exists a ...
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43 views

Open balls in the definition of a Euclidean submanifold

Stroock (Essentials of Integration Theory For Analysis, $\S8.3.4$) defines a $k$-dimensional submanifold of $\mathbb{R}^n$ to be a set $M \subseteq \mathbb{R}^n$ such that for all $x \in M$, there ...
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52 views

The Product map of a Lie Group is a Submersion.

Problem 7.1 of Lee's Introduction to Smooth Manifolds (2nd Edition) reads: Show that for a Lie group $G$, the multiplication map $\mu:G\times G\to G$ is a submersion (Hint: Use Local Sections). ...
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38 views

Looking for a reference that explains connections and curvature by double tangent space

I'm looking for a book or a set of lecture notes on differential manifolds that explain connections (Levi-Cevita connection, prinicipal connections) and curvature on an abstract manifold from the ...
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26 views

Computing the differential of multiplication by $M$ on $U(n)/O(n)$

Suppose $M\in U(n)$. Then multiplication by $M$ induces a smooth action on $U(n)/O(n)$. How can we compute the differential of this map? If $M$ were acting on a matrix group, then of course the ...
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49 views

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ bilinear … so is it tensor like?

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ linear in both components... so is it a tensor of some kind? I know (I think) it is not a ...
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44 views

Transformation laws for tensors on general manifolds

I was interested in tensor products rather from the mathematical, abstract point of view, so topics like various tensor products in the context of, for example, Banach spaces, $C^*$-algebras and so ...
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30 views

Zeroes of differential form embedded

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ ...
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28 views

Decomposition of tangent space of principal bundle

A connection on a principal bundle $\pi:P\rightarrow M$ is a choice of horizontal subspace $H_p$ at each $p\in P$, such that $T_p P = H_p + V_p$ where $V_p = \ker((\pi_*)_p)$. It is very common to ...
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57 views

Question for experts in dynamical systems or symplectic geometry

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof): In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...
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38 views

Taylor series representation for a Riemannian hypersurface

This is Exercise 8.5 in Lee's Riemannian Manifolds: An Introduction to Curvature. Suppose $M\subset \mathbb R^{n+1}$ is a hypersurface with the induced metric. Let $p\in M$, and let ...
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49 views

Natural operators in differential geometry?

Which operators in differential geometry is called natural? And this neutrality is respect to what property or structure? Why this is an important problem? and what is due problem relations with lie ...
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48 views

Representation of conjugate directions

Is there a way to represent conjugate directions on a Mohr circle of curvature? ( Surface Theory, Second fundamental form, M = 0 ) Directions given by double angles AOB, AOC. Is this attempt ...
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62 views

“Maximum point lies on a curve” implies tangential derivative is zero there.

Given a differentiable function $f:\mathbb{R}^2\to\mathbb{R}$, suppose that it has a local maximum at the point $(x_0,y_0)$. Let $\gamma$ be a smooth curve passing through $(x_0,y_0)$. Does it follow ...
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81 views

Differential geometry for nonlinear control theory

I am engineering student and I need to acquire a good understanding of some notions in differential geometry such as manifold, diffeomorphism, distributions etc.But I can't find a proper starting ...
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17 views

Relation between torsion in torsion free of covariant derivative and torsion free group

Is there a relationship between "torsion free" of covariant derivatives and the torsion free group? Or is this just coincidence that people use the term "torsion free" here? It is in general required ...
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52 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
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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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31 views

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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23 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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28 views

$g_{ij}$ 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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19 views

Topologically non trivial cycles

I am studying a Stiefel manifold $X$ which is topologically an $S^3$ bundle over an $S^4$ but is not a product space. I am not able to understand that why is it not the product space $S^3 \times S^4$? ...
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26 views

Reciprocal relations in Roulette /glissette rollings

If a catenary rolls on a straight line its focus traces out a parabola and vice versa. Is it true? Are there more such examples and how are they co-related? In case of a circle rolling on a fixed ...
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37 views

Consequences of the Carathéodory conjecture

This is a very stupid question. What are consequences and applications of the Carathéodory conjecture? It seems to me interesting, but completely useless.
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34 views

$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
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Hermitian metric on line bundle over the Grassmannian

We know that the Grassmannian manifold $G(k,\mathbb{C}^n)$ can be embedded in the projective space $\mathbb{C}P^N$ for $N= {n\choose k}-1$ by the Plucker embedding $P$. On $\mathbb{C}P^N$ we have ...
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45 views

Line integral and differential forms

Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$ I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is ...
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35 views

Huisken's distance comparison principle and type II singularities.

I've been reading Huisken's paper on his distance comparison principle and he remarked that in particular his theorem rules out the formation of type II singularities. These are singularities where in ...
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23 views

Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
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27 views

Curvature shortening flow of embedded curves

QUESTION: I'm not sure how they proved part c in particular. Note that theorem 2.1 refers to Huiskan's distance comparison principle for evolving curves. I don't see why a separating boundary curve ...
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75 views

Difference between exponential maps composed with parallel transport along two different geodesics?

Let $(M,g)$ be a Riemannian manifold, and let $\gamma_{p,v}, \gamma_{p,w}$ be two geodesics starting from $p$ with directions/initial vectors $v,w$ respectively. Consider the two operations (to be ...
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41 views

The curvature blows up

A curve evolves according to the evolution equation $\displaystyle\frac{\partial X}{\partial t}= k \times N$ where $k$ is the curvature of the curve and $N$ is the inward unit normal vector. Then ...
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45 views

Given an embedded $C^k$ submanifold of $R^n$, is the distance function $C^k$?

I am asking because I am wondering if studying smooth manifolds is the same as studying zero loci of smooth functions with non-vanishing gradient. Let me be a little formal for clarity: Let $M$ be a ...
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25 views

Criteria for boundary convexity of hypersurfaces in Euclidean space

I have a question about the relationship between two different formulations of the notion of boundary convexity, in the sense of Riemannian geomety. Let $M$ be an $n$-dimensional manifold with ...
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56 views

Connection giving an identification of the double-tangent bundle

Let $M$ be a manifold, and let $\nabla$ be a (possibly torsion-free) connection on $M$. Then I am pretty sure that $\nabla$ induces an isomorphism between the double-tangent bundle and the Whitney sum ...
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37 views

$\alpha(t_0)$ orthogonal to $\alpha^\prime (t_0)$

Somebody asked this question a while back but only hints were posted--I want to have my entire proof examined. Let $\alpha(t)$ be a parametrized curve which does not pass through the origin. If ...
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32 views

Is the dimension of a smooth manifold an invariant of the underlying set in it?

Let $M$ be a smooth manifold, $S$ a set and $f:M\to S$ a bijection (assuming of course, that such a function does exist). It's an easy exercise to show that $S$ can be given a differentiable ...
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63 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
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34 views

Exercise about Hodge Star Operator on $\Lambda^{p,q}$

In real case, $$ \ast (e_1\cdots e_k)=e_{k+1}\cdots e_n$$ on $\mathbb{R}^n$, where $e_i$ is $1$-form and $e_1\cdots e_n$ is volume on $\mathbb{R}^n$ We will extend to complex case. Define $$ dz_k:= ...
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61 views

Problem of Arnold's book and covering spaces

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
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34 views

Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
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46 views

Kinds of isometries preserving the curvature tensor

We talked about isometries /local isometries and linear isometries in the lecture, but unfortunately we did not say when the isometry preserves the curvature tensor or sectional curvature. First I am ...