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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How to find the equation of the normal line to the surface S

How to find the equation of the normal line to the surface $S$: $$f(u,v)=(2u-v,u^2+v^2,u^3-v^3)$$ at the point $M(3,5,7)$? Could someone post the complete solution?
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1answer
52 views

Calculate the curvature of a space curve at the point M(-1,5,-4)

In order to calculate the curvature for this space curve, do I use this formula? And where does the point $M$ come into this? P.S. This is probably a silly question, but I'm new to differential ...
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2answers
177 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
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1answer
78 views

Question about the second fundamental form

I am studying Riemannian geometry and have a question understanding something. I use Do Carmo's book. In the book, a vector field is defined for isometric immersions: for an immersion $$ ...
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1answer
43 views

Cocycles vector bundles and metrics

It is well known, and not difficult to prove that a vector bundle $E$ over a (smooth) manifold $M$ together with a metric gives rise to orthonormal frames (by Gram-Schmidt). An consequnece is that the ...
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89 views

Radius of curvature for the plane curve $x^3 + y^3 = 12xy$.

Could someone help me with this problem? : Determine the radius of curvature for the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$.
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1answer
86 views

Reference request-What is the prerequisite of S.S.Chern's proof of the generalised Gauss-Bonnet theorem?

The title basically explains everything. The OP is an independent learner, who in the current stage sets S.S.Chern's proof of the generalised Gauss-Bonnet theorem as the goal. But what is the ...
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2answers
649 views

Area of a weird ellipse shape.

A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ ...
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2answers
58 views

Why do we need three indices for Christoffel Symbols

I read the following results on covariant differentiation (summation convention applied): If $X=X^i e_i$ and $Y=Y^je_j$, then $$\nabla_XY = X^i\nabla_{e_i}(Y^je_j) = X^ie_i(Y^j)e_j + ...
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1answer
78 views

Intuition/visualization for a non-flat connection

I'd just like to check whether my visualization for a way to get a non-flat connection is correct. The definition I am using for a connection is, for a fiber bundle $\rho:E \to B$, a smooth ...
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1answer
25 views

Why this doesn't transform properly?

We are in $\mathbb R^n $, with a tensor field of components $T_\nu$, and being $e_\mu$ the vectors of the basis: $e_\mu \equiv \partial_\mu$, then I'm asked to show that $\partial_\mu T_\nu$ can't ...
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117 views

Recover Covariant Derivative from Parallel Transport

It is well known that one can recover the connection from the parallel transport. I struggle to understand this concept. Since $\Gamma(\gamma)^t_s:E_{\gamma(s)}\to E_{\gamma(t)}$ is an isomorphism ...
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168 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
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1answer
31 views

Find the derived of an implicit given function.

Let $C=\{(x,y,z) \in \mathbb(R)^3| \sin x + \sin^2 y + \sin^4 z=0 \ \text{and} \ (x-z)^2=4\pi^2\}.$ By the implicit function theorem, we have that $C$ can be parametrized as a smooth curve in the ...
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73 views

Tangent developable of helix.

Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$. 1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$. 2) I want to find how many times a line can ...
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2answers
127 views

How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array} $$ are $\mathbb{C}$-linearly independent, even if the underlying ...
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3answers
106 views

Why is $d*F$ equal to $\partial _\mu F^{\mu \nu}$?

Given that $A = A_\nu dx^\nu$ and $F = \partial_{\mu}A_\nu dx^\mu \wedge dx^\nu$ Why does $d*F$ equal to $\partial _\mu F^{\mu \nu}$? How does all the $\frac{1}{2}\varepsilon^{abcd}F_{cd}$ fit into ...
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1answer
40 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
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1answer
15 views

mapping between differential forms and its property

I am trying to prove the following property of the map between differential forms: (Spivak's book ''Calculus on manifolds'' p.91) $$f^{\star}\;\Lambda^{k}(\mathbb{R}^{m}_{f(p)})\to ...
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0answers
21 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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1answer
26 views

Vectors that span normal bundle of a submanifold

If $E_{1}=\frac{\partial}{\partial y}-kx\frac{\partial}{\partial w}$ and $E_{2}=cos\psi\frac{\partial}{\partial x} + sin \psi\frac{\partial}{\partial w}$ span tangent space of $M$, where $M$ is a ...
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2answers
39 views

Gauss's theorem in 2d: how can it be expressed in differential forms?

How do we express the 2d version of Gauss's theorem in the language of differential forms? In 3d, I know it is $$d \left(Fdydz + Gdzdx + Hdxdy\right) = F_x + G_y + H_z dxdydz$$ so by Stokes' ...
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138 views

Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$, i.e., each ...
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139 views

Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$. ...
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1answer
51 views

If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$

I'm having the same problem as it was questioned here. I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n ...
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1answer
33 views

Question on Normal Coordinates

I'm having a hard time trying to understand something that I'm suspicious is pretty stupid. I'll refer to Wikipedia to settle the term's I'll refer to. ...
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1answer
64 views

Multi Index Dirac delta function

If we assume the following result: $$\delta^{\alpha_1,\alpha_2,\cdots , \alpha_k, \rho}_{\beta_1,\beta_2,\cdots , \beta_k, \rho} = (n-k)\delta^{\alpha_1,\alpha_2,\cdots , ...
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1answer
32 views

Tensorproduct of vectorbundles

Assume we got $\pi:E \to M$, a vector bundle of manifolds. I know how to make the bundle $(E \otimes E)^* \to M$. But how do the local trivializations look like ? I suppose that if ...
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1answer
75 views

Why the geodesic curvature is invariant under isometric transformations?

As I know the geodesic curvature $$ \kappa_g = \sqrt{det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ ...
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Linear transformation from endormorphism to real number

For a finite dimensional vector space $V$, is there a linear transformation between its endomorphism and real number, please? I suspect that since the element of the endomorphism can be represented by ...
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2answers
63 views

Trace of $\nabla \alpha$ where $\alpha$ is a 1-form.

Let $X$ be a Riemannian manifold, $\nabla$ be a connexion and $\alpha$ be a 1-form. How do I show that $\text{Tr}(\nabla\alpha)=\sum e_{i}\alpha(e_{i})-\alpha(\nabla_{e_{i}}e_{i})$ (where $e_{i}$'s ...
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building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
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Why are 'differential operators on manifolds' differential operators?

It is clear what is meant by a differential operator on $\mathbb{R}^n$ (or some open subset). However, it is not clear to me why differential operators on smooth manifolds are defined the way they ...
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0answers
44 views

Invariants of shape operators?

Let $S:V\longrightarrow V$ be a Linear Transformation, then the Characteristic polynomial of $S$ and therefore its Coefficient are invariant. Except the first and the last Coefficient that we know ...
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1answer
100 views

Prove the Gaussian curvature $K=0$

If two families of a geodesics on a surface intersect at a constant angle $\theta$, prove that the Gaussian curvature of the surface is zero, i.e. $K=0$. Please explain how to show the question. ...
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25 views

Curvature of the boundary curve of convex set

I've got a very simple question, but I can't get a rigorous proof. Suppose that $X\subset\mathbb{R}^2$ is a convex set, and suppose further that $\gamma$ is a closed regular curve with image $\partial ...
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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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1answer
35 views

If $G$ is a compact Lie group acting as bundle automorphisms of $L$ then why can we choose the metric on $M$ and the connection to be $G$-invariant.

Let $M$ be a compact Manifold and $L\to M$ be a Hermitian Line bundle compatible with connection $\nabla$ and $G$ is a compact Lie group acting as bundle automorphisms of $L$ then why can we choose ...
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1answer
127 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
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1answer
70 views

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
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1answer
33 views

Differential of a smooth function

I have seen that if $f$ is a smooth function on a smooth manifold $M$ then differential of $f$ at the point $p$ is defined by $(df)_p(X_p) = X_p(f)$ but i am not able to see that how it can be derived ...
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1answer
54 views

Prove: If $\int_{\phi}\omega = \int_{\psi}\omega$ whenever $\phi$ and $\psi$ have the same endpoints, then $\omega=df$

This is an exercise that I have been assigned for homework. I don't really know how to approach it though. I know that $\int_{\phi}\omega$ only depends on the endpoints $\phi(a)$ and $\phi(b)$ where ...
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Clarification: What does it mean when “$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points”

"$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points" Does this mean: (A) $\phi:[a,b]\to U$ and $\psi:[a,b]\to U$ (B) $\phi:[a,b]\to U$ and $\psi:[c,d]\to U$ s.t. ...
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1answer
136 views

Another differential topology lemma

Another lemma (1) Why can we assume $z=f(z)=0$ and that $U$ is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ...
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1answer
58 views

Exponential Map: Diffeomorphism on Normal Balls

Consider a differentiable manifold depicted with an affine connection. How is the exponential map constructed (sketch)? What if given rather a bundle and a connection?
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Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
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2answers
73 views

Motivation of vector bundle of a manifold

I am studying about vector bundle from M.Lee but not getting the feel of it. Can someone explain me about the importance of vector bundle? Why do we need to study about it? Thanks! Also, i have to ...
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1answer
57 views

Equivalent Definition of a Hermitian Metric on an Almost Complex Manifold?

For an almost-complex manifold $M$ with almost-complex structure $J$, we say that a metric $h:T_p(M;{\mathbb R}) \times T_p(M;{\mathbb R}) \to {\mathbb R}$ is Hermitian if it holds that $$ ...
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1answer
62 views

Proving that something is a manifold

I'm a beginner at differential geometry and I'm having some trouble with the following problem: Let $M \subset \mathbb{R}^n$ be a $k$-dimensional smooth manifold (smoothly embedded in ...
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1answer
34 views

A detailed question in Bott&Tu

In Bott&Tu:《Differential Forms in Algebraic Topology》 p.26,there is a statement: (b) $\Omega_{c}^{\ast}$ is a covariant functor under inclusions of open sets Obviously for any two arbitary open ...