The Gaussian curvatures of spheres

1. An example of the Gaussian curvature

Example 1 Compute the Gaussian curvature of sphere

\displaystyle S = \{(x, y, z) \in \mathbb{R}^3 | x^2 + y^2 + z^2 = R^2\}.

Parametrizing {X:}

\displaystyle U \subset \mathbb{R}^2 \rightarrow S

\displaystyle (u, v)\mapsto (u, v, \sqrt{R^2 - u^2 - v^2}).

we have

\displaystyle X_u = (1, 0, -\frac{u}{\sqrt{R^2 - u^2 - v^2}}), X_v = (0, 1, -\frac{v}{\sqrt{R^2 - u^2 - v^2}})

The coefficients of the second fundamental form:

\displaystyle E = \langle X_u, X_u \rangle = 1 + \frac{u^2}{R^2 - u^2 - v^2}

\displaystyle G= \langle X_v, X_v \rangle = 1 + \frac{v^2}{R^2 - u^2 - v^2}

\displaystyle F = \langle X_u, X_v \rangle = \frac{uv}{R^2 - u^2 - v^2}

\displaystyle \Rightarrow EG - F^2 = (1 + \frac{u^2}{R^2 - u^2 - v^2})(1 + \frac{v^2}{R^2 - u^2 - v^2}) - \frac{u^2v^2}{(R^2 - u^2 - v^2)^2}

\displaystyle = 1 + \frac{u^2}{R^2 - u^2 - v^2}+ \frac{v^2}{R^2 - u^2 - v^2}

\displaystyle EG - F^2= \frac{R^2}{R^2 - u^2 - v^2}

we compute coefficients of the first fundamental form:

\displaystyle N= -\frac{X_u \wedge X_v}{\|X_u \wedge X_v\|}

\displaystyle X_u \wedge X_v = (\frac{u}{R^2 - u^2 - v^2}, \frac{v}{R^2 - u^2 - v^2}, 1)

\displaystyle \|X_u \wedge X_v\| = \sqrt{\frac{R^2}{R^2 - u^2 - v^2}}

\displaystyle = \frac{R}{\sqrt{R^2 - u^2 - v^2}}

\displaystyle \Rightarrow N = -\frac{X_u \wedge X_v}{\|X_u \wedge X_v\|} = -(\frac{u}{R}, \frac{v}{R}, \frac{\sqrt{R^2 - u^2 - v^2}}{R})

\displaystyle X_{uu} = (0, 0, - \frac{R^2 - v^2}{\sqrt{(R^2 - u^2 - v^2)^3}})

\displaystyle X_{vv} = (0, 0, -\frac{R^2-u^2}{\sqrt{(R^2-u^2-v^2)^3}})

\displaystyle e = \langle N, X_{uu} \rangle = \frac{v^2 - R^2}{R}\cdot \frac{1}{R^2 - u^2 - v^2}

\displaystyle g = \langle N, X_{vv} \rangle = \frac{u^2 - R^2}{R}\cdot \frac{1}{R^2 - u^2 - v^2}

\displaystyle f = - \langle N_u, X_v \rangle = \frac{uv}{R(R^2 - u^2 - v^2)}


\displaystyle N_u = (-\frac{1}{R}, 0, \frac{u}{R\sqrt{R^2 - u^2 - v^2}})

\displaystyle X_v = (0, 1, -\frac{v}{\sqrt{R^2 - u^2 - v^2}})

These imply that

\displaystyle eg - f^2 = \frac{1}{R^2(R^2 - u^2 - v^2)^2}[(u^2 - R^2)(v^2 - R^2) - u^2v^2]

\displaystyle = \frac{1}{R^2(R^2 - u^2 - v^2)^2}[- (u^2 + v^2)R^2 + R^4] = \frac{R^2[R^2 - u^2 - v^2]}{R^2(R^2 - u^2 - v^2)^2}

\displaystyle = \frac{1}{R^2 - u^2 - v^2}.

By the above computation, the curvature of sphere is

\displaystyle K = \frac{eg - f^2}{EG - F^2} = \frac{1}{R^2}.


MATH-F-420: Differential geometry of Verbitsky

Misha Verbitsky

Université Libre de Bruxelles

MATH-F-420: Differential geometry

Monday 16:00-18:00, P.OF.2058

Announcement for this course.



Miscellanea: test problems, exam, etc.



On the diffetential of a mapping 2

In the case {\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m} case, there is a linear map, which is “linear approximation” of {\psi}. In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If {M} and {N} are smooth manifolds and {\psi \colon M \rightarrow N} is a smooth map then for each {m \in M}, the map

\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N

is defined by

\displaystyle d\psi(v)(f) = v(f \circ \psi)

is called the pushforward. Actually,

\displaystyle d\psi \colon TM \rightarrow TN.

Suppose that {\dim{M} \ge \dim{N}} and {f \colon M \rightarrow N} is a differentiable mapping. We have

Definition 1 The mapping {f} is called a trivial fibration (differentiable) on {N} if there exists a differential manifold {F}, is called fibre of {f}, and a diffeomorphism

\displaystyle \phi \colon M \rightarrow N \times F

such that the following diagram is commutative


Một ví dụ về mặt chính quy

BT. Cho hàm f(x,y,z)=z^{2}. CMR 0  không là giá trị chính qui của hàm f thế nhưng f^{-1}(0) vẫn là mặt chính qui.

Lời giải. 

Ta có ma trận (f_x \ f_y \ f_z) = (0 \ 0 \ 2z) suy biến tức rankA < 1 khi và chỉ khi z=0, do đó tại điểm (x, y, 0), \forall x, y \in \mathbb{R}, ma trận trên suy biến và do đó là điểm kì dị. Mà f(x,y,0)=0 nên 0 là giá trị tới hạn, hay ko phải giá trị chính qui. Nhưng f^{-1}(0) là mặt phẳng Oxy, đây là mặt trơn nên chính qui.

The tangent plane of a surface and the tangent space of a manifold (at one point)

The tangent plane of a surface
By a tangent vector to S at a point p \in S, we mean the tangent vector \alpha'(0) of a differentiable parametrized curve \alpha: (-\epsilon, \epsilon) \to S with \alpha(0) = p.

What is the tangent plane of a surface? That is a plane which containes all of the tangent vectors of this surface at point p \in S.

Let \varphi: U \subset \mathbb{R}^2 \to S be a parametrization of a regular surface S and let q \in U. The vector subspace of dim 2,

d\varphi_q(\mathbb{R}^2) \subset \mathbb{R}^3,

coincides with the set of tangent vectors to S at \varphi(q)

There is a similar situation in here. In the case of manifold, the tangent space is also built from the set of all of tangent vectors of a manifold.

Tangent spaces of a manifold

In Milnor’s book (\cite{Milnor}), the tangent space TM_x at x for arbitrary smooth manifold M \subset \mathbb{R}^k is defined:

Choose a parametrization g : U \to M \subset \mathbb{R}^k of a neighborhood g(U) of x in M, with g(u) = x. We have dg_u: \mathbb{R}^m \to \mathbb{R}^k. So the image dg_u(\mathbb{R}^m) of dg_u is equal to TM_x.


J. W. Milnor, Topology from differentiable viewpoint, 1965.

M. do Carmo, Differential geometry, curves and surfaces, 1976.