The Weingarten map of S at p is the linear map
L_p : S_p \to S_p
where L_p(v) = - \nabla_vN. L_p measures the rate of change of N as it passes through p along \alpha
Measure the rate of change of the unit normal vector N to the surface
f: U \to \mathbb R, \alpha: I \to U, and v \in \mathbb R_p^{n+1} then
\nabla_v f = (f \circ \alpha)'(t_0)
where \alpha(t_0) = p and \dot\alpha(t_0) = v
When ||v|| = 1 it is called the directional derivative.
Properties:
Let X be a smooth vector field on an open set U \subset \mathbb R^{n+1} with respect to v \in \mathbb R_p^{n+1}, p \in U:
\nabla_vX = \dot{(X \circ \alpha)}(t_0)
where \alpha: I \to U, \alpha(t_0) = p and \dot\alpha(t_0) = v
Let N be an orientation of S. Then \nabla_vN is tangent to S.
When X is a tangent vector field on S then D_vX is the tangential component of \nabla_vX:
D_v X = \nabla_v X - (\nabla_vX \cdot N(p))N(p)
Let S be an n-surface in \mathbb R^{n+ 1}, oriented by the unit normal vector field N. Let p \in S and v \in S_p. Then for every parametrized curve \alpha: I \to S with \dot \alpha(t_0) = v for some t_0 \in I,
\ddot \alpha(t_0) \cdot N(p) = L_p(v) \cdot v
The Weingarten map is self-adjoint, that is
L_p(v) \cdot w = v \cdot L_p(w)
for all v, w \in S_p
Let U be an open subset of \mathbb R^2 and f: U \to \mathbb R be a smooth function. A plane curve is a 1-surface C oriented by \frac{\nabla f}{||\nabla f||}
The weingarten l_p: C_p \to C_p map is a linear transform on a 1-dimensional space. Let e_1 be the basis of C_p and L_p(e_1) = me_1. Then \forall_{v\in C_p} v = a_1 e_1, for some a_1 \in \mathbb R
\kappa(p) = \frac{L_p(v) \cdot v}{||v||^2}, \kappa(\alpha(t)) = \frac{\ddot \alpha(t)\cdot N(\alpha(t))}{||\dot\alpha(t)||^2} is the curvature in \mathbb R^2.
Thus L_p(v) = \kappa(p)v
If:
A parametrization of a segment of a plane curve C containing p \in C is a parametrized curve \alpha: I \to C which:
If Im(\alpha) = C then it is a global parametrization.
Let \alpha: I \to \mathbb R^{n+1} be a parametrized curve with \dot \alpha(t) \ne 0 for all t \in I then there exists a unit speed parametrization \beta of \alpha.
For a plane curve C with non-zero curvature \kappa there exists a unique oriented circle O at each p \in C which
The length of a parametrized curve \alpha: I \to \mathbb R^{n+1} is
L(\alpha) = \int_a^b ||\dot \alpha(t)|| dt
For a, b \in I. Unit speed curves are parametrized by arc length, that is L(\alpha) = b - a (when ||\dot \alpha|| = 1)
If C is an oriented plane curve then there exists a global parametrization of C iff C is connected.
If C is a connected oriented plane curve and \beta: I \to C is a unit speed global parametrization of C then, \beta is either one to one or periodic.
\beta is periodic iff C is compact
If V is a vector space over the field K, then a linear map T: V \to K is called a form
Let V be a vector space over a field F. Then the dual of V is the space V^* of linear maps T: V \to F, T_1, T_2 \in V^*.
A differential 1-form on an open set U \subset \mathbb R^{n+1} is a function \omega: U \times \mathbb R^{n+1} \to \mathbb R such that for all p \in U the restriction \omega | p: \{p\} \times \mathbb R^{n+1}_p \to \mathbb R is a linear map.
\omega_X(p, v) = X(p) \cdot (p, v) is a 1-form dual to X
Let f: U \to \mathbb R be a smooth function. Then the differential of f is the 1-form:
df(v) = \nabla_vf
Let x_i: U \to \mathbb R where x_i(a_1, a_2, \cdots, a_{n+1}) = a_i. Then its differential is dx_i = v_i.
Given a vector field X and a 1-form \omega we define a function \omega(X): U \to \mathbb R
\omega(X)(p) = \omega(X(p))
For every 1-form \omega on U \subset \mathbb R^{n+1} there exist unique functions f_i: U \to \mathbb R such that
\omega = \sum_{i=1}^{n+1} f_i dx_i
Let U \subset \mathbb R^{n+1} and f: U \to \mathbb R be smooth them,
df = \sum_{i=1}^{n+1} \frac{\partial f}{\partial x_i} dx_i
The integral of an exact 1-form over a closed connected oriented curve is always equal to zero.
Let \alpha: [a,b] \to U be a parametrized curve. We define the integral of a 1-form \omega over \alpha as
\int_\alpha \omega = \int_a^b \omega(\dot\alpha(t))dt
Let \eta be the 1-form on \mathbb R^2 \setminus \{0\} defined by
\eta = -\frac{x_2}{x_1^2 + x_2^2}dx_1 + \frac{x_1}{x_1^2 + x_2^2}dx_2
Then for a closed, piecewise smooth, parametrized curve \alpha: [a, b] \to \mathbb R^2 \setminus \{0\} we have
\int_\alpha \eta = 2\pi k
For some integer k. This number is called the winding number of alpha about the origin.
Normal curvature of S at p in the direction of a unit length vector v is the number k(v) = L_p(v) \cdot v
The normal section determined by a unit vector \vec v = (p, v) \in S_p where p \in S is the subset \mathcal N(v) of \mathbb R^{n+1} given by
\mathcal N(v) = \{q \in \mathbb R^{n+1} : q = p + xv + yN(p) \text{ for some } (x, y) \in \mathbb R^2\}
Or equivalently, a map
i: \mathbb R^2 \to \mathbb R^{n+1}, (x, y) \mapsto p + xv + yN(p)
There exists an open set V containing p such that S \cap \mathcal N(v) \cap V is a plane curve.
The curvature of at p of this curve is equal to the normal curvature k(v).
Let V be a finite dimensional vector space with a dot product and let L: V \to V be a self-adjoint linear transformation on V. Let S = \{v \in V: ||v|| = 1\} and define f: S \to \mathbb R by f(v) = L(v) \cdot v. Suppose f is stationary at v_0 \in S (v_0 is a critical point of f). Then L(v_0) = f(v_0)v_0. So v_0 is an eigenvector of L with eigenvalue f(v_0).
Let V be a finite dimensional vector space with a dot product and let L: V \to V be a self-adjoint linear transformation on V. Then there exists an orthonormal basis for V consisting of eigenvectors of L.
\mathcal L_p(v) = L_p(v) \cdot v
When ||v|| = 1 then \mathcal L_p is the normal curvatures
On each compact oriented n-surface S in \mathbb R^{n+1} there exists a point p such that the second fundamental form at p is definite.