Recent content by lavinia

In terms of the sequence, SO(2)->SO(3)->S^2, since SO(2) is closed in SO(3) ,this sequence describes a fiber bundle whose fibers are cosets of some fixed copy of SO(2). I guess from this approach one next asks which circle bundle over S^2 this is. Under the action of SO(3) on the 2 sphere's...

I was asking because I wanted to visualize the inability to have a null homotopy. I am thinking of the case of the 3 sphere where all closed geodesics are null homotopic as are the closed geodesics in its tangent sphere bundle.

Ok. But how does one know that these critical geodesics are not-null homotopic from this?

@Matterwave The first description in Post #9 of the tangent circle bundle of the 2 sphere as a circlular stack of of projective planes that share a common circle can be intuitively thought of as a book whose "binding" is the shared circle and whose pages are the projective planes. An easier...

@wrobel can you explain that a little more. I gather that the geodesic here is the full rotation around an axis. And the Riemannian structure is the one that SO(3) inherits from its action on the tangent circle bundle of the sphere?

yes. The tangent space to the sphere at any point is just a two dimensional plane so the vectors of length 1 in each plane make a circle centered at the origin. Geometrically, one can think of the tangent bundle as a subset of R^3xR^3 as all points (x,v) where x is a point on the sphere and v...

The Tangent Circle Bundle of the 2-Sphere Here is a way to “visualize” the tangent circle bundle of the 2-sphere that connects to a lot of rich topology. Start with a smooth vector field, V, that has a single isolated zero at the north pole of the sphere. This vector field has index 2 which...

A mathematician showed me another way to see the 3 sphere. Slice the 3 sphere along its equatorial 2 sphere. Its two hemispheres are 3 dimensional balls which can be seen by projecting them vertically into three dimensions. This is the same as slicing a regular sphere in three dimensions. If...

To expand @wroblel's point in post #3: The action of SO(3) on the 2 sphere can be extended to an action on its tangent circle bundle by the mapping r:(x.v)-> (r(x),dr(v)) where r is a rotation of the sphere. This mapping defines a diffeomorphism of SO(3) onto the tangent circle bundle of the 2...

@WWGD I thought you'd like see see why evolutes are critical values of the normal map. For a curve in the plane, the normal map is c(s) + tN(s) where N(s) is a unit normal to the curve c(s). If c(s) is parameterized by arc length and N(s) points in the direction of c'' , a classical equation...

I think for hypersurfaces of Euclidean n-space one gets n-1 evolute like hypersurfaces as the critical values of the normal mapping. This just generalizes the case of a curve in the plane. In the hypersurface case there are specific curves called principal curvatures along which the normal...

For the parabola (x,x^2) plugging into the formula for the evolute gives the parametric equation (-4x^3,1/2+3x^2). This is not a parabola because of the cubic term. I thought perhaps an inductive proof would show that the parametric equation for the n'th evolute would be a pair of polynomials...

i tried to indicate a method of proof in reply #4

The evolute of a parabola contains a cubic term so is not a parabola. I wonder if the degree of iterated evolutes of the parabola is increasing.

One intuitive idea of defining a differentiable structure on a triangulated manifold is to embed the manifold in Euclidean space and then smooth out the vertices and edges to eliminate all of the creases and sharp points. For instance a tetrahedron can be made into a smoothly embedded sphere by...