This is an expression I came across in a paper I am going through. It involves an expression for the parallel transport of a tangent vector taking into consideration the sectional curvature of simply connected space-forms in \mathbb R^4 . I have not been able to derive it.The equation simply...
cone as surface...
In Barrett O'Neill's Elementary Differential Geometry book...he says that the cone
M:x^{2} + y^{2}=z^{2} is not a surface in that
there exists a point p in M such that there exists no proper patch in M which can cover a neighbourhood of p in M
Intuitively I realize that...
first of all thanks a ton both of u...JG89 and Bacle...
@JG89...i know very little about patches ...i am using Barrett O'Neill...but the content of ur post is pretty clear...however can this be done without bringing in patches??...
also..by Bacle's statement...shudnt it be
dx\left\langle...
I am unable to understand as to how the basis for the tangent space is
\frac{\partial}{\partial x_{i}}. Can this be proved ,atleast intuitively?
Bachman's Forms book says that if co-ordinates of a point "p" in plane P are (x,y), then
\frac{d(x+t,y)}{dt}=\left\langle 1,0\right\rangle...