# Questions tagged [orientation]

For question regarding the notion of orientation both in topology and in global analysis.

287 questions
50 views

### If $\phi$ is a orientation preserving linear automorphism, do we really need to choose the same orientation for the domain and the codomain?

In the book of Linear Algebra by Werner Greub, at pages $131-132$, it is given that and However, when $\phi$ is a linear automorphism, it says that $\Delta_F = \Delta_E$, and derives the following ...
21 views

### Question related to orientation on a arbitrary oriented manifold

This is a section from Loring Tu's book Introduction to Manifolds page 244 Second Edition. My question is as follows: Towards the end of the text in the image he says that an oriented manifold can be ...
17 views

### On the unit sphere $S^2$, show the antipodal map $A:S^2\to S^2$ is orientation reversing using definitions.

I want to show that on the unit sphere $S^2$, the antipodal map $A:S^2\to S^2$ given by $(x,y,z) \mapsto (-x,-y,-z)$ is orientation reversing. I know that $S^2$ is a regular connected orientable ...
25 views

### Pairings of Simplex Edges

This is an open-ended question. If you'll allow, I'd like to keep its origins vague for the moment, so as not to bias responses. I am interested in any and all thoughts. There are three pairings of ...
13 views

77 views

### Conditions for Matrix to be Product of Near-Identity Matrices

For $\epsilon > 0$, let $M_{\epsilon}$ be the family of $n$ x $n$ real matrices A such that $||$A$-$I$_n|| < \epsilon$, where $|| \cdot ||$ is the standard operator norm. If $\epsilon$ is ...
987 views

### What is the difference between the orientation and the direction of a vector?

I came across this recently, and was confused regarding the difference between the orientation and the direction of a vector. Does the orientation refer to the relative coordinate which the vector is ...
56 views

### Orientation of a Manifold with Trivial Tangential Bundle

Let $M$ be a smooth (eg $C^{\infty}$) manifold. Let assume that $M$ has trivial, oriented tangent bundle $TM$, so $TM \cong M \times \mathbb{R}^n$ for appropriate $n$ and orientable. How to conclude ...
24 views

### Finding and Comparing 2 Sensors Rotations, with same reference frame but different initial Orientation

Let's say we want to Compare two different Arm (Humerus) Rotations (series of quaternions) and we do not care about space translation but only for rotation. To measure each rotation we use the same ...
937 views

### How to rotate an orientation (Euler angles)

If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the ...
1k views

### How to calculate rotation quaternion between two orientation quaternions?

I have some device (3D pointer) connected to my computer which returns it's position (in cartesian XYZ system) and orientation (in quaternions). I receive this values about 30 times/sec. Now I need ...
30 views

### The orientation induced on the boundary of a manifold.

I just learned about the notion of orientability of a manifold which is difficult and abstract for me. If we consider all basis of a vector space, the matrix that transforms one basis in another basis ...
675 views

### Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this ...
12 views

81 views

### Orientation in the proof of Stokes Theorem

I'm reading the proof of Stokes theorem at page 83 of "Godinho, Natàrio, An introduction to Riemannian geometry" and I can't understand a passage in it, probably because the definition of ...
32 views

### Find the angles between two solids

I have 2 solids (A and B) and I need to find the three angles between their x, y, and z axes. If I calculate the geometrical center of the two solids (Ax, Ay, Az and Bx, By, Bz), is it correct to ...
40 views

### Prove there exists a outward unit normal field on the boundary this manifold

Let $M$ be a compact subset of $\mathbb{R}^3$ with the standard orientation $\mu =[e_1,e_2,e_3]$ and let $S = \partial{M}$ is its smooth boundary with the induced orientation from $M$. Prove there ...
50 views

### Are closed (topological) submanifold in $\mathbb R^n$ of codimension 1 orientable?
See Are closed, properly embedded manifolds of co-dimension 1 in $\mathbb{R}^n$ orientable? for treatment of the smooth case. If the topological case of Jordan theorem holds for such manifolds, ...
### $df_x$ maps $N(S.X)$ isomorphically
This is actually just a linear algebra problem, but this text is taken from Guillemin and Pollack, Differential Topology on page 100. Real Problem: Let $f: X \to Y$ be smooth, $S = f^{-1}(Z)$ where ...