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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96 views

Convex polyhedron and its Gauß-curvature

I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. What we know: Gauß-Curvature at every vertex is given by $K(p) = 2\pi - \sum\limits_{\text{angle } ...
3
votes
2answers
41 views

Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
2
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1answer
20 views

a curve does not need to be injective?

In diff. Geometry, curve is a differentiable mapping from an open interval to 3 dimensional euclidean space. Doesn't it need to be injective? If it is not, then there might be a two different tangent ...
2
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1answer
35 views

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$ It is easy for me to check that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $2$ using the following theorem: Let $F:U ...
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1answer
28 views

Calculate the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...
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1answer
43 views

Is this a sufficient condition for differentiability

Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that, for all $c\in \mathbb{R}$, every vector in $f^{-1}(c)$ is supported by a unique hyperplane to $f^{-1}(c)$. Is $f$ ...
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vote
1answer
24 views

Congruence of two curves with an arbitrary speed?

I'm studying the book "elementary differential geometry" by o'neil. There is a collorary which states that if two curves a(t), b(t) which is defined in the same real line interval has the same speed, ...
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1answer
33 views

Can someone explain whether $C^{\infty}(M)$ is an algebra or a commutative ring?

One forms $\Omega^1(M)$ is a module over $C^{\infty}(M)$, therefore does that make $C^{\infty}(M)$ an algebra or a commutative ring?
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vote
1answer
43 views

Find an atlas for $H=\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4 : x_1+x_2^2=x_3^2+x_4=1\}$

Find an atlas for $H=\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4 : x_1+x_2^2=x_3^2+x_4=1\}$ Let $F:\mathbb{R}^4 \rightarrow \mathbb{R}^2$ s.t. $(x_1,x_2,x_3,x_4) \mapsto (x_1+x_2^2-1,x_3^2+x_4-1)$. ...
4
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0answers
36 views

Twisted geodesics on circular torus

I am attempting to make a Mechanics of materials approach to describe non-linear deformations of thin lines/wires on a torus. A mathematical modeling of its probable geometry is required at start of ...
3
votes
0answers
31 views

Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} ...
8
votes
1answer
39 views

Embedding covers of manifolds

I am considering $k$-fold covers of smooth manifolds (with smooth covering maps). Let $f:M^m\to N^m$ be a smooth finite covering map. -- The following implication is not true: $M$ can be embedded ...
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0answers
16 views

Showing tractrix revolution is isometric to hyperbolic plane

I have a tractrix defined (in the xy plane) by the fact that the line segment formed by a tangent to the curve meeting the x axis has length 1. Given that the tangent meets the x axis at an angle ...
2
votes
1answer
46 views

$C^\infty$ bump function to smoothen a corner

Let $\beta(t)$ be a smoothener at $t=0,$ e.g. $\beta(t)=e^{-1/t},$ for $t\in \mathbb{R}^+.$ Let's say that this is a horizontal smoothener, as it flattens at $\beta(0^+)$. Now I want another function ...
1
vote
1answer
45 views

Finding incomplete geodesics

I have a problem with the notion of incomplete geodesics. Can someone give me a minimal example for such a geodesic? In particular, I am trying to solve the following exercise: Consider the upper ...
4
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0answers
36 views

Manifolds with 'bad metrics' (reference request)

While studying some differential geometry, a thought crossed my mind that I am sure has been considered before, but I cannot find a reference for it. What can be said about spaces for which the ...
0
votes
1answer
37 views

Find the curvature

If a point moves along a curve so that the velocity and acceletation vectors have constant lenght, how to proove that the curvature is also a constant?
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0answers
20 views

comparing models of spherical geometry

There more than two models of spherical geometry? 1) one for the half sphere (taking the north pole as center, the boundary is the equator) 2) one for the whole sphere (taking the north pole as ...
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0answers
20 views

Constructing an immersion of a curve with certain properties

Consider the map $\gamma : \mathbb R \to \mathbb R^2$ given by $(t^2, t^3)$. Let $Gr(1,T\mathbb R^2) := \bigcup_{x \in \mathbb R^2} Gr(1,T_x \mathbb R^2)$ where $Gr(1,T_x \mathbb R^2)$ is the ...
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vote
1answer
56 views

Curvature and Circumference of Circle

Theorem Let $\gamma\colon [a,b]\rightarrow \mathbb{R}^2$ be a unit speed simple closed curve, with $\gamma'(a)=\gamma'(b)$ and $N$ is the inward-pointing normal. Then $$ \int_{a}^b ...
2
votes
0answers
42 views

What is meant by the discriminant locus of a fibration?

If we have a fibration $f: X \to B$ (allowing singular fibers) of a differentiable manifold $X$, precisely what is meant by "the discriminant locus of $f$ " $\Delta \subset B$ and how do we define it? ...
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1answer
71 views

Integration by parts formula on unbounded manifold

Let $M$ be a closed Riemannian manifold and set $X = M \times [0,\infty)$ with the trivial product metric induced. If $u$ and $v$ are functions defined on $X$, how do I know that the formula $$\int_X ...
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1answer
38 views

$e^{xy}dx \wedge dy$: determine the $1$-form that it induces on $S^1$ and check if the obtained $1$-form respects or not the induced orientation

Consider the $2$-form $e^{xy}dx \wedge dy$ on $\mathbb{R}^2$. Determine the $1$-form that it induces on $S^1$, viewed as the boundary of $B_2$. Check if the obtained $1$-form respects or ...
6
votes
1answer
47 views

Making a set into a manifold

Let $n \in \mathbb{N}$, $M$ be a set and let $\mathcal{A} = \{(\varphi_a, U_a)\}_{a \in \mathcal{A}}$ be a system of tuples so that: $U_{a} \subseteq \mathbb{R}^n$ is open for all $a$; $\varphi_a: ...
1
vote
1answer
31 views

Show that $M=\{(x,y,u,v) \in \mathbb{R}^4 : x^2 + y^2 = u^2 + v^2 = 1\}$ is orientable, explaining the induced orientation.

Let $M=\{(x,y,u,v) \in \mathbb{R}^4 : x^2 + y^2 = u^2 + v^2 = 1\}$. Show that $M$ is an orientable subvariety of $\mathbb{R}^4$, explaining the induced orientation. Consider the $2$-form ...
0
votes
1answer
23 views

Confusion on when components of a vector relative to a basis are not components of a tensor

I have been studying affine connections, parallel transport and the covariant derivative. The text I am reading defines an affine connection $\nabla$ as a map $\nabla ...
2
votes
1answer
27 views

An inequality for absolute total curvature in Riemannian surfaces

Let be $M\subseteq \mathbb{R}^3$ a compact (Riemannian) surface and let be $K$ the gaussian curvature of $M$. I want to prove that $$ \int_{M} |K| \geq 4\pi(1+g(M))$$ where $g(M)$ is the genus of ...
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0answers
37 views

Is this the obvious immersion or is it a mistake in this example?

Consider the map $$ f: \mathbb R^2 \to \mathbb R^3, \hspace{0.5cm} (t^2 + 2s, t^3 +3ts , t^4 + 4t^2 s)$$ The goal of the following example is to find an immersion $\widetilde{f}: \mathbb R^2 \to ...
1
vote
1answer
64 views

Ricci tensor and average of a tensor

Let $(M^n,g)$ be an oriented Riemannian $n$- manifold and $g$ is a Riemannian metric on $M$ , $\mathrm{d}\sigma$ is Riemannian volume form on $S^{n-1}$ and $\text{Vol}(S^{n-1})$ is volume of ...
5
votes
0answers
38 views

Dimensions of Grassmannians?

I'm trying to work out the dimensions of some examples of Grassmannians but I can't seem to do it. Here is what I understand: The Grassmanian $G(k,n)$ is the set of all $k$ planes in $\mathbb ...
0
votes
1answer
36 views

Regular values on boundary of smooth manifold

Let $X$ be a smooth compact manifold without boundary and $Y$ be a smooth compact manifold with boundary $\delta Y$ where $\dim X = \dim Y$. Suppose $ f: X \rightarrow Y $ is smooth. As shown in ...
2
votes
0answers
24 views

Parametrization admitting conservation of K & H

Which mappings admit conservation of K and H ? ( Gauss and mean curvatures). Apart from helicoid/catenoid isometry, which examples can be given of surface bending and distortion so that mean ...
2
votes
0answers
29 views

a problem on geometry of hypersurfaces

Recently I am reading book on mean curvature flow by carlo mantegazza.There I found a problem on hypersurfaces stated below : Show that if the hypersurface $M \subset {R}^{n+1}$ is locally the graph ...
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0answers
22 views

Can a bubble be modelled convincingly using a ricci flow?

My knowledge of this type of geometry leaves a lot to be desired, but I have heard about the Ricci flow. My understanding of the Ricci flow is that it models a "velocity" for each part of a shape as ...
2
votes
4answers
101 views

Notation of a function in coordinates

I have a question about some notation which puzzles me a lot. Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$. Then people often write or say that if we choose coordinates $x=(x^1,…,x^n)$ ...
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vote
1answer
35 views

Isometry in Hyperbolic space

Let $\mathbb{H}^2=\{ (x,y)\in\mathbb{R}|\ y>0 \}$ the hyperbolic space with the metric $g=(dx^2+dy^2)/y^2$. Let ...
0
votes
1answer
16 views

Vector-valued function to describe a hyberboloid

I need to find a vector-valued function to describe the quadric surface $x^2+y^2-z^2=1$. I could use the identity $\cosh^2 u - \sinh^2 u = 1$, but I'm not sure how. The best I could arrive at is ...
1
vote
0answers
45 views

Interior Derivative and Contraction: Kobayashi and Nomizu.

In Kobayashi & Nomizu, the interior derivative of an r-form is defined as $\iota_X \omega = C(X \otimes \omega)$, where $C$ is the contraction associated with the pair $(1,1)$ and $\omega$ is ...
2
votes
2answers
63 views

Lie group step in proof

Let $X_e,Y_e \in T_eG$ be vectors and $G = GL(n).$ Then the right translation is given by $Y_g = Y_e g$ and $X_g = X_e g.$ Now, I have a proof showing that $[X_e,Y_e] \in T_eG$ is the element ...
2
votes
1answer
25 views

Finding distance from point to line which is perpendicular to another line

Find the distance of the point $(1,1,1)$ from $x+y+z=1$ measured perpendicular to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{6}$
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vote
1answer
44 views

Differential geometry related question

This question is a follow-up to a question posted here After tweaking the parametrization a little bit, I decided to go with this: $$x=\frac{1}{2}\cos(t) , y=\frac{1}{2}+\frac{1}{2}\sin(t) , ...
2
votes
1answer
27 views

Choose the reflection planes of a surface through a single point.

Let $S$ be a surface in $R^3$, for which coordinate vector field of $S$ has zero mean on $S$. Assume that for any vector $n$, a normal plane to $n$ exist, such that $S$ is symmetric about it. How can ...
4
votes
1answer
46 views

Coordinates on the sphere not global?

I'm reading a book on differential geometry and some part of the introduction I do not understand but I'm curious to understand it. Maybe someone can try to explain those parts to me. "Each point on ...
0
votes
0answers
18 views

Shape of generalized helix

This is a homework assignment that I fail to understand. The problem is to find the shape of a generalized helix $r=r(s)$ when the fixed vector is $v=(0,0,1)$ and $r(0)=0$. I have found forms that ...
1
vote
1answer
16 views

Example of a surjective submersion that is not locally a product?

A surjective submersion $p:E \rightarrow M$ which is proper (compact sets of M have compact preimages) over connected $M$ is a fibre bundle ie has local products with a standard fibre. If we drop ...
4
votes
1answer
44 views

Find the parametric equation of the curve to be the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $y=z$

A space-curve $C$ is defined to be the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $y=z$. How should one try to find the parametric equation of the curve? It seems natural to let ...
2
votes
1answer
34 views

Indices at the left of a tensor in mathematical physics/differential geometry?

I am a mathematician and I am reading a paper in mathematical physics and I found the following notation: Let $Y$ be a two–form on $M$ such that $$\nabla({}_iY_j)_k = 0.$$ Here, $\nabla$ is ...
2
votes
2answers
33 views

$(e^x \cos y,e^x \sin y)$ is local diffeomorphism not global

Let $f:\mathbb{R^2} \to \mathbb{R^2}$ defined by $f(x,y)=(e^x \cos y,e^x \sin y)$ I have showed that $f$ is a local diffeomorphism by using inverse function theorem, that is $\det(Df)=e^x \gt 0$ for ...
0
votes
1answer
51 views

Self-adjoint extension of the Laplacian

Let $M$ be a complete Riemannian manifold and $-\Delta$ denote the Laplace-Beltrami operator on $M$. We can prove that $(-\Delta f, g) = (\nabla f, \nabla g) = (f, -\Delta g)$, when $f, g \in ...
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votes
0answers
20 views

Is the hyperbolic paraboloid a minimal surface?

Assuming the boundary to be the appropriate 4 face diagonals of a cube, is the hyperbolic paraboloid a minimal surface?