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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253 views

Background for reading Milnor's Morse Theory book

I wish to study the book 'Morse Theory' by J. Milnor, but I am not sure whether I have the necessary prerequisites. I know basic point set topology, real analysis (limits, continuity, differentiation, ...
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186 views

Curvature parametrized by arc length

Suppose $\alpha$ is a curve parametrized by arc length and there is some $s_0$ such that $||\alpha(s)||\le ||\alpha(s_0)||$, $\forall s$ near $s_0$. Show that: $$\kappa(s_0) \ge ...
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1answer
43 views

$\lambda : (x,y,x^2+\lambda y^2)$ contains straight lines

For which $\lambda\in \mathbb{R}$ $$\varphi(x,y) = (x,y,x^2+\lambda y^2)$$ is a ruled surface, i.e. it can also be parametrized as $$ \mathbb{R}^2 \ni (t,u) \mapsto g(t) + uw(t) $$ with $g$ a ...
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81 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
3
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1answer
166 views

Isometry in $\mathbb{R}^2$

Will there be an isometry in $\mathbb{R}^2$ taking the curve $\alpha(t)=(\cos(t)+1, \sin(t)+2)$, where $t\in [0,\pi]$, to the curve $\beta (t)=(t,\sin(t))$, where $t\in[0,c]$ and $c$ is a ...
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1answer
52 views

Is there a Tangent Vector that will Trace the Position Vector?

My calculus book featured several graphing problems that involved drawing the position vector and the tangent vector at a certain point on the graph of a vector equation. Trying to trace several of ...
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42 views

For $k \times n$ matrices $X,Y$, when do we have $X = AY$ for $A \in \operatorname{Gl}(k, \mathbb{R})$?

Let $\mathcal{M}_{k,n}(\mathbb{R})$ denote the set of $k \times n$ matrices with entries in $\mathbb{R}$. Let $\operatorname{Gl}(k, \mathbb{R})$ denote the set of all $k \times k$ invertible matrices ...
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57 views

Immersion on a compact set

Let $K\subset \mathbb R^n$ be compact and convex. Let $U$ be an open neighborhood of $K$ and $f : U\rightarrow \mathbb R^m$ an injective immersion. Must there exist constants $a,b >0$ such that if ...
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28 views

meaning of divergence and its mathematical intuition [duplicate]

how does divergence which means sink or source equal to ∂Fx/∂x+∂Fy/∂y +∂Fz/∂z.I have been thinking it for a long time and i think "divergence tells us how fast the vector increases when we move apart ...
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168 views

real meaning of divergence and its mathematical intuition

how does divergence which means sink or source equal to ∂Fx/∂x+∂Fy/∂y +∂Fz/∂z.I have been thinking it for a long time and i think "divergence tells us how fast the vector increases when we move apart ...
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2answers
40 views

Fencing with only area of $3$ square feet

If I had $6$ feet of fencing could I fence a region that has area $3$ square feet? So, I must show that there is a curve in the plane of my fencing that has length $6$ feet that bounds the ...
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1answer
206 views

Values of the Christoffel symbols

Are the values of the christoffel symbols the same for all coordinate systems on a surface/manifold? I would love to see an example for the cone in two different parametrizations.
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1answer
156 views

Can the diagonal of a manifold be expressed as the zero set of a section of a vector bundle?

Let $\mathbb{C} \mathbb{P}^2$ be the two dimensional complex projective space and $$M:= \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.$$ Let $ \gamma \rightarrow \mathbb{C} \mathbb{P}^2 $ be ...
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853 views

Tangent and Normal lines that pass through the origin

Let $\alpha$ be a regular curve in $\mathbb{R}^2$ and let all of its tangent lines pass through the origin. Also, let $\beta$ be a regular curve in $\mathbb{R}^2$ and let all of its normal ...
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1answer
76 views

An oriented curve

An oriented curve has a unique arc length parameterization inducing the given orientation. If I let $C$ be a curve parameterized by arc length by $\alpha: [0, p]\to \mathbb{R}^2$ how can I ...
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39 views

Sufficient statistics and isometries

Let $(M,g)$ be an infinite dimensional statistical manifold with the Fisher information metric $g$. Is it true that any isometry on this manifold must correspond to a sufficient statistic?
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162 views

Isometry Group of a Manifold

Let $(M,g)$ be a Riemannian manifold and let $I = Iso(M)$ be the group of isometries of $M$. Suppose that $I$ has no subgroups. What does this tell us about $M$?
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489 views

Show the Grassmannian is a smooth manifold (using dummy definition of smooth manifold)

We received the following problem in my Differential Geometry class: Suppose $0\leq k \leq n$ are integers. Let $G(k,n)$ be the collection of orthogonal projections $T: \mathbb{R}^n \to \mathbb{R}^n$ ...
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369 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
7
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1answer
210 views

What's the intuition behind the tangent bundle?

Well, when we work with a smooth manifold $M$ we can associate with each point $p\in M$ a vector space $T_p M$ of all vectors at $p$ tangent to $M$: this is the space of linear functionals obeying ...
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156 views

The Ricci flow and $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$ are equivalent up to diffeomorphism

Suppose $M$ is a Riemannian manifold. Consider flow $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$, where $f$ is a time-dependent function. I would like to prove that flows of this ...
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249 views

What do the curvature and torsion measure? [duplicate]

Consider a smooth surface for simplicity. What does its curvature measure? What does its Gaussian/Riemannian curvature measure? What does its torsion measure? What does the Ricci curvature measure?
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60 views

The connection on the principal bundle

Suppose $M$ is a manifold, and $E$ a vector bundle over $M$ equipped with a connection $\nabla $. If $F$ is the frame bundle of $E$, is there an explicit construnction of a connection on $F$ ...
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56 views

Does anyone know this Formula and a Proof/Context (Possibly-related to Cartan's Formula)

I'm Trying to understand the formula; for $w$ a 2-form with d the exterior derivative , and $X,Y$ vector fields; $[X,Y]$ is the Lie bracket: $dw(X,Y)=X(w(Y))-Y(w(X))-w([X,Y])$ I was hoping someone ...
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1answer
59 views

What functions on the plane (and on $\mathbb{R}^n$) have projection-valued derivatives?

Thinking about a more general problem I am trying to work out a specific case: If $U\subset \mathbb{R}^2$ is a connected open set what are the differentiable (or $C^r$) functions $f\colon ...
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1answer
88 views

Divergence theorem: why is it $F_n \cdot dS$?

I am learning the calculus 3 interpretation of Stokes' theorems while just dipping my toe in the water of differential forms. Since I have only limited knowledge of the real differential geometry ...
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1answer
97 views

Products of differential forms

Let us consider $\mathbb R^n$. Let $\{x_i\}$ be a basis. Let $\{dx_i\}$ be the corresponding basis in the double dual. Do we assign meaning to the symbol $dx_idx_j$, and what does it mean? What do ...
0
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1answer
87 views

Standard notation for isometry group?

Let $M$ be a (semi) Riemannian manifold, is there a standard notation for the group of isometries on $M$? I would think $\mathrm{ISO}(M)$ would be appropriate, but I've never encountered a dedicated ...
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1answer
72 views

Derivative with respect to the Frenet frame

Given a regular smooth curve $\alpha: I \to \mathbb{R}^3$, we have that the Frenet Frame $\vec{t},\vec{n},\vec{b}$ is an orthonormal basis of $\mathbb{R}^3$ at each point $s \in I$, and hence we can ...
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246 views

Unit Tangent Vector and Unit Normal Vector

I was trying to compute the unit normal vector of $\alpha(t)=(cos^3(t),sin^3(t))$ and was finding that it gets particularly unpleasant after the curve is reparameterized by unit length. Are there any ...
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62 views

Christoffel Symbol Not Disappearing

If I am given a vector field $\vec{A}(x,y) = x^2 \hat{e}_1 + y^2 \hat{e}_2 = (A^x,A^y)$, I'd like to calculate it's covariant derivative in the $r$ direction after expressing the vector field in polar ...
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202 views

If a smooth manifold X is covered by an odd sphere, then X is orientable.

In solving some old qualifying exam questions, I've been thoroughly stumped. If a smooth manifold $X$ is covered by an odd dimensional sphere, then $X$ is orientable. I see this question has ...
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1answer
128 views

The equation of a surface created by the extrusion of a 2D closed curve along a path

How do I obtain the equation of a surface created by the extrusion of a circle (or ellipse) created on the XY plane along a parabola or a parametric curve which lies on the YZ plane. The goal is to ...
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3answers
889 views

How can we show the cone $x^2 +y^2 = z^2$ is not a smooth manifold?

In our differential geometry class, as a preliminary we have used as the definition of a manifold the following: $M \subset \mathbb{R}^n$ is a $k$-manifold if for each point $p\in M$ there exists a ...
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1answer
510 views

Is a connected sum of manifolds uniquely defined?

It is a standard excercise in differential geometry to prove that a connected sum $M\#N$ of two smooth manifolds $M,N$ of the same dimension is uniquely defined (under some assumptions regarding ...
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1answer
125 views

How to induce a connection on endomorphism bundle?

Assume $\nabla:C^\infty(E) \rightarrow C^\infty(T^*X \otimes E)$ is a covariant derivative and $u$ is an element in the endomorphism bundle $End E$. I'm confused why is the induced connection of ...
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2answers
108 views

Translating french paper into English

I am currently studying a french paper on Einstein manifolds by Berard Bergery and I have doubts that my translation of the following sentence is correct: "De plus, puisque $G$ agit par isometries, ...
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1answer
94 views

What is 'target manifold'?

I saw in a lecture about O(3) sigma model something about 'target manifold', but I do not know what is it. Is there any book I could learn about that?
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1answer
63 views

Way distinguishing whether or not complex manifold

$SU(3)$ has dimension 8. Why is this not a complex manifold ? Thank you in advance.
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1answer
65 views

Trajectory of circumference [circle] rolling down any given curve

How should i go about describing mathematically the path traced by the center of a circumference [circle] rolling down (or up) any given curve described by $y = f(x)$? The solution for a linear ...
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1answer
133 views

showing that a local diffeomorphism is a local isometry using first fundamental form

In differential geometry, there is a theorem about 1st fundamental form : A local diffeomorphism $f:S_1 \rightarrow S_2$ is a local isometry $\Leftrightarrow$ For any patch $\sigma$ of $S_1$, ...
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1answer
40 views

Meaning of CR-Automorphism

What is the meaning of the CR-Automorphism and CR-Manifold? I tried to find the definition from the web. Is it Continuous Real ....? Thanks.
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85 views

Problem about finding Gaussian curvature

The problem is about Gaussian Curvature When we define $M=r(u,v)$ be the surface parametrized by $(u,v)$ in $\mathbb{R}^3$ and $N$ be a unit normal vector of $M$. In my lecture we denote Gaussian ...
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1answer
70 views

curves and constant length

Find all curves in $\mathbb{R}^2$ having the following property: the segment of the normal straight line between curve and the x axis has constant length. If $\alpha (t)=(x(t),y(t))$, I found ...
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1answer
85 views

Finding all alternating bilinear $T$ that preserve a certain group of isometries of $\mathbb{R}^{n+1}$

Let $$G=\left\{\begin{pmatrix} H & 0 \\ 0 & 1\end{pmatrix} \ | \ H\in O(n), HJ=JH \right\}\subset \mathrm{Lin}(\mathbb{R}^{n+1},\mathbb{R}^{n+1}) $$ where: $n=2m$, $J$ is the standard complex ...
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77 views

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...
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1answer
340 views

Question in Do Carmo 1-2

In Manfredo Do Carmo's Differential Geometry of Curves and Surfaces, Section 1-2, he asks: Let $\alpha: I \to \mathbb{R}^3$ be a smooth curve that does not pass through the origin.If $\alpha(t_0) ...
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2answers
174 views

How can I compute the area of a geodesic triangle?

How can I compute the area of a geodesic triangle in a Riemannian 2-manifold? If the Gauss curvature $K$ is constant and positive I can take the Gauss-Bonnet theorem to obtain ...
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1answer
84 views

Confused by local isometries

I think I am a little confused about the notion of a local isometry of Riemannian manifolds. Let's say I have a manifold $(M,g)$ where $g$ is the Riemannian metric. Take a chart $x:U \rightarrow ...
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1answer
115 views

Transport theory basics: can't understand solid angles

I don't understand something in transport theory: $$P(x,\vec{w})=p(x,\vec{w}) \cos\theta \, dw \, dA$$ This is the number of particles flowing across a differential surface element in the direction ...