# Second Fundamental Form

**Second Fundamental Form** - Web like the rst fundamental form, the second fundamental form is a symmetric bilinear form on each tangent space of a surface. Web the second fundamental form is a function of u = u1 and v = u2. Note that nu and nv are both orthogonal to n, and so lie in the tangent. Tp(σ) ×tp(σ) → r k: Web so the second fundamental form is 2 1+4u2+4v2 p (du2+dv2): (3.29) and , , are called second fundamental form coefficients. Unlike the rst, it need not be positive de nite. T p ( σ) × t p ( σ) → r is given through the weingarten map χ χ, i.e. Web the second fundamental form on the other hand encodes the information about how the surface is embedded into the surrounding three dimensional space—explicitly it tells. Then we have a map n:m → s2 n:

(1.9) since ei;j = ej;i, the second fundamental form is symmetric in its two indices. Therefore the normal curvature is given by. Also, since we have x12 ~ = x21, ~ it follows that l12 = l21 and so (lij) is a symmetric matrix. Web for a submanifold l ⊂ m, and vector fields x,x′ tangent to l, the second fundamental form α (x,x′) takes values in the normal bundle, and is given by. It is a kind of derivative of the unit. The second fundamental form is given explicitly by. Web the extrinsic curvature or second fundamental form of the hypersurface σ is defined by.

Web the numerator of ( 3.26) is the second fundamental form , i.e. Unlike the rst, it need not be positive de nite. E = ii p(x u;x u);f = ii p(x u;x v);g = ii p(x v;x v): (53) exercise1.does this mean at anypointp2s, the normal curvature nis a constantin everydirection?. Therefore the normal curvature is given by.

It is a kind of derivative of the unit. Unlike the rst, it need not be positive de nite. (3.29) and , , are called second fundamental form coefficients. Together with the first fundamental form, it serves to. Therefore the normal curvature is given by. Web the second fundamental form is a function of u = u1 and v = u2.

Web the coe cients of the second fundamental form e;f ;g at p are de ned as: Unlike the rst, it need not be positive de nite. (53) exercise1.does this mean at anypointp2s, the normal curvature nis a constantin everydirection?. Having defined the gauss map of an oriented immersed hypersurface,. Web it is called the second fundamental form, and we will term it bij:

Web the second fundamental form describes how curved the embedding is, in other words, how the surface is located in the ambient space. Web the numerator of ( 3.26) is the second fundamental form , i.e. Web it is called the second fundamental form, and we will term it bij: Web the coe cients of the second fundamental form e;f ;g at p are de ned as:

### Web The Second Fundamental Form K:

Then we have a map n:m → s2 n: The second fundamental form is given explicitly by. (3.29) and , , are called second fundamental form coefficients. Web different from the first fundamental forms, which encode the intrinsic geometry of a surface, the second fundamental form encodes the extrinsic curvature of a surface embedded.

### Extrinsic Curvature Is Symmetric Tensor, I.e., Kab = Kba.

U ⊂ ir3 → ir be a smooth function defined on an open subset of ir3. (u, v) ↦ −u ⋅ χ(v) ( u, v) ↦ − u ⋅ χ ( v). Web like the rst fundamental form, the second fundamental form is a symmetric bilinear form on each tangent space of a surface. The quadratic form in the differentials of the coordinates on the surface which characterizes the local structure of the surface in a.

### Web The Second Fundamental Form On The Other Hand Encodes The Information About How The Surface Is Embedded Into The Surrounding Three Dimensional Space—Explicitly It Tells.

Also, since we have x12 ~ = x21, ~ it follows that l12 = l21 and so (lij) is a symmetric matrix. Web so the second fundamental form is 2 1+4u2+4v2 p (du2+dv2): ( p) is a unit vector in r3 ℝ 3, it may be considered as a point on the sphere s2 ⊂r3 s 2 ⊂ ℝ 3. It is a kind of derivative of the unit.

### Web It Is Called The Second Fundamental Form, And We Will Term It Bij:

$$ \alpha (x,x') = \pi. Web another interpretation allows us to view the second fundamental form in terms of variation of normals. Fix p ∈ u and x ∈ tpir3. Web the fundamental forms of a surface characterize the basic intrinsic properties of the surface and the way it is located in space in a neighbourhood of a given point;