Is $SO(3)$ Path-Connected?

  • MHB
  • Thread starter Euge
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In summary, $SO(3)$ is the special orthogonal group in three-dimensional Euclidean space, representing all rotations that preserve the origin. It is path-connected, meaning there is a continuous path between any two elements, making it useful in fields such as computer graphics and robotics. Proving its path-connectedness can be done using the concept of homotopy. $SO(3)$ also has other interesting properties, such as being a compact group, having a non-trivial center, and being isomorphic to $SU(2)$. It has various applications in fields such as quantum mechanics and computer vision.
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Euge
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Here is this week's problem!

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Show that $SO(3)$, the space of all real $3\times 3$ orthogonal matrices of determinant $1$, is path-connected.
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  • #2
No one answered this problem. You can read my solution below.

Fix $M\in SO(3)$. There is a factorization $M = QA(\theta)Q^T$ where $Q$ is orthogonal, $A(\theta) = 1 \oplus R(\theta)$ and $R(\theta))$ is a two-dimensional rotation matrix. The map $F : [0,1] \to SO(3)$ defined by $F(t) = QA(t\theta)Q^T$ is a continuous path from the identity matrix to $M$. Hence, $SO(3)$ is path-connected.
 

FAQ: Is $SO(3)$ Path-Connected?

1. What is $SO(3)$?

$SO(3)$ is the special orthogonal group in three dimensions, also known as the rotation group in three-dimensional space. It is the set of all 3x3 orthogonal matrices with determinant 1. In simpler terms, it is the group of all possible rotations in 3D space.

2. What does it mean for a group to be path-connected?

A group is path-connected if there exists a continuous path between any two elements in the group. This means that you can continuously transform one element into another without leaving the group. In other words, there are no "gaps" or "jumps" in the group.

3. How do we determine if $SO(3)$ is path-connected?

We can determine if $SO(3)$ is path-connected by showing that any two elements in the group can be continuously transformed into each other. This can be done by explicitly constructing a path between the two elements or by using a topological argument.

4. Why is it important to know if $SO(3)$ is path-connected?

Knowing if $SO(3)$ is path-connected is important because it tells us about the structure and properties of the group. It also has implications in various fields, such as physics and computer graphics, where rotations in 3D space are commonly used.

5. What is the significance of the answer to "Is $SO(3)$ Path-Connected?"

The answer to this question has important implications in mathematics and its applications. If $SO(3)$ is path-connected, it means that the group has a simple and connected structure, which can aid in understanding its properties and applications. On the other hand, if $SO(3)$ is not path-connected, it would indicate a more complex and possibly fragmented structure, which may have different implications in different contexts.

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