Sign of geodesic curvature

With the usual notation we have $ \chi=1 $ let us consider for a spherical cap in $ \mathbb R^3$ the theorem

$ $ \int K dA + \int \kappa_{g} \,ds= 2 \pi \chi = 2 \pi $ $

For a polar cap when a polar latitude circle is drawn infinitesimally near North Pole, the respective terms are

$ $ 0 + 2\pi =2 \pi$ $

For a hemisphere cap when an equator /great circle is drawn in the center

$ $ 2\pi +0 = 2 \pi$ $

For a polar cap when a polar latitude circle is drawn infinitesimally near South Pole enclosing all of the sphere surface

$ $ 4\pi -2 \pi =2 \pi$ $

Now considering same differential geometric definition of the same circle how do we obtain $ \int \kappa_{g} \,ds = \pm 1$ for North pole or South Pole?

Curvature of projection function onto a smooth curve

Suppose we have a smooth curve $ C$ lying in $ \mathbb{R}^2$ , and let us consider the orthogonal projection function $ P_C(x)$ onto the curve, described by $ $ P_C(x) = argmin_{y \in C} \Vert x – y \Vert$ $ where $ \Vert \cdot \Vert$ is a norm, it can be $ \Vert \cdot \Vert_2^2$ , or $ \Vert \cdot \Vert_1$ .

My question is: is there a general relationship between the second derivative tensor of $ P_C(x)$ and the curvature of the curve $ C$ ? For example, relationship between their norms, and if the curve is convex, what can we say about the second derivative of the projection? If nothing specific can be said, under what restrictions on the curve $ C$ and/or location of $ x$ can we say something about their relationships? Does there exist work that discusses this problem or some problems related to it?

To visualize the problem somewhat, we consider an example here: Example

Denoting the blue curve as $ C_1$ and black curve as $ C_2$ , $ C_1$ clearly has greater curvature than $ C_2$ , but what about $ \Vert D^2P_{C_1}(x) \Vert$ vs. $ \Vert D^2P_{C_2}(x) \Vert$ ?

determinant of curvature (notation issue)

This is when studying about Chern classes from Kobayashi and Nomizu.

Let $ \pi:E\rightarrow M$ be a complex vector bundle with fibre $ \mathbb{C}^r$ and Group $ G=GL(r,\mathbb{C})$ .

Let $ p:P\rightarrow M$ be associated principal $ G$ bundle. Let $ \mathfrak{g}=\mathfrak{gl}(r,\mathbb{C})$ denote the Lie algebra of $ G$ .

Given $ B\in \mathfrak{g}$ , the determinant $ \text{det}\left(\lambda I_r-\frac{1}{2\pi\sqrt{-1}} B\right)$ is $ \sum_{k=0}^r a_k\lambda^{r-k}$ for some $ a_k\in \mathbb{C}$ .

Given $ B\in \mathfrak{g}$ , we have $ r$ elements in $ \mathbb{C}$ . Varying $ B$ over $ \mathfrak{g}$ , gives $ r$ functions $ f_k:\mathfrak{g}\rightarrow \mathbb{C}$ .

We have $ $ \text{det}\left(\lambda I_r-\frac{1}{2\pi\sqrt{-1}} X\right)=\sum_{k=0}^r f_k(X) \lambda ^{r-k}$ $ These $ f_k:\mathfrak{gl}(r,\mathbb{C})\rightarrow \mathbb{C}$ are homogeneous, degree $ k$ polynomial functions on $ \mathfrak{gl}(r,\mathbb{C})$ . I can recall what are polynomial functions on a vector space if some one needs it.

These are $ GL(r,\mathbb{C})$ invariant i.e., $ f_k(X)=f_k(DXD^{-1})$ for all $ D\in Gl(r,\mathbb{C})$ . These $ f_k$ gives a symmetric, multilinear, $ Gl(r,\mathbb{C})$ invariant mappings $ f_k\in I^k(G)$ .

Let $ \Gamma$ be a connection on $ P(M,G)$ and $ \Omega$ be its curvature form. This $ f_k$ gives a $ 2k$ form $ f_k(\Omega)$ on $ P$ . Let $ \gamma_k$ be the unique closed $ 2k$ -form on $ M$ such that $ p^*(\gamma_k)=f_k(\Omega)$ .

We then have $ $ \sum_{k=0}^r f_k(\Omega)=\sum_{k=0}^rp^*(\gamma_k)=p^*(1+\gamma_1+\cdots+\cdots+\gamma_r)$ $ . Then, they (Kobayashi and Nomizu, page no $ 307$ ) write $ $ \text{det}\left(I_r-\frac{1}{2\pi \sqrt{-1}} \Omega\right)= p^*(1+\gamma_1+\cdots+\cdots+\gamma_r)$ $ . I see that they are just replacing $ X$ with $ \Omega$ . But what does it mean to say determinant of $ \Omega$ ?

Calculating curvature of a contour

I have a equation of a scalar field in the form

f(x)=x^2+y^2+xy+c

I want to find the curvature of the contour of the curve at fc=f(0.5,0.5).

So I need to calculate the derivative dy/dx and d/dx(dy/dx)

I can solve the equation f(x,y)=fc and get the derivative of f(x,y) w.r.t x

On paper we do,

d/dx (f(x,y))=d/dx(fc) 2*x+2*y*(dy/dx)+y+x(dy/dx)=0 (dy/dx)=-(2*x+y)/(x+2*y) 

and further d/dx(dy/dx) for a curvature approximate

how can I rearrange the equation such that I can get the value of dy/dx on mathematica