12.3 Submanifolds and induced geometry

Submanifolds are like mini-worlds within bigger spaces. They inherit their structure from the larger space but have their own unique features. This section dives into how these smaller worlds fit into the bigger picture.

We'll look at how submanifolds get their shape and measurements from the larger space. We'll also explore special tools, like the Gauss map, that help us understand how these mini-worlds curve and bend in the bigger space.

Submanifolds and Induced Geometry

Defining Submanifolds and Their Properties

• Submanifold refers to a subset of a larger manifold with its own manifold structure
• Inherits smooth structure and topology from the ambient manifold
• Can be defined locally as the zero set of smooth functions (implicit definition)
• Alternatively described using a parametrization or embedding (explicit definition)
• Dimension of a submanifold is always less than or equal to the dimension of the ambient manifold
• Classified as regular or immersed based on the properties of the inclusion map

Induced Metric and First Fundamental Form

• Induced metric arises naturally on a submanifold from the metric of the ambient manifold
• Represents the restriction of the ambient metric to the tangent spaces of the submanifold
• Mathematically expressed as $g_{\text{induced}}(X, Y) = g_{\text{ambient}}(dF(X), dF(Y))$ where $F$ is the inclusion map
• First fundamental form encodes the induced metric in local coordinates
• Denoted as $I = g_{ij} dx^i dx^j$ where $g_{ij}$ are the components of the induced metric
• Determines intrinsic geometry of the submanifold (lengths, angles, areas)

Isometric Embeddings and Their Significance

• Isometric embedding preserves the metric structure when mapping one manifold into another
• Satisfies the condition $\langle X, Y \rangle_M = \langle dF(X), dF(Y) \rangle_N$ for all tangent vectors $X$ and $Y$
• Crucial in studying the relationship between intrinsic and extrinsic geometry
• Nash embedding theorem guarantees the existence of isometric embeddings for Riemannian manifolds into Euclidean spaces
• Applications include visualizing abstract manifolds and analyzing their geometric properties

Normal Bundle and Gauss Map

Normal Bundle Structure and Properties

• Normal bundle consists of all vectors orthogonal to the tangent spaces of a submanifold
• Defined as $N_pM = \{v \in T_pN : \langle v, w \rangle = 0 \text{ for all } w \in T_pM\}$ at each point $p$
• Complement to the tangent bundle in the ambient tangent space
• Dimension equals the codimension of the submanifold in the ambient manifold
• Smooth vector bundle structure inherited from the ambient manifold
• Crucial in studying the extrinsic geometry of submanifolds

Gauss Map and Its Geometric Interpretation

• Gauss map assigns to each point on a submanifold its unit normal vector
• For hypersurfaces, maps into the unit sphere of the ambient space
• Formally defined as $G: M \to S^n, p \mapsto \nu(p)$ where $\nu$ is a unit normal vector field
• Measures how the submanifold curves in the ambient space
• Differential of the Gauss map relates to the shape operator
• Generalizes to higher codimension using the Grassmannian manifold

Unit Normal Vector and Its Role

• Unit normal vector represents the direction perpendicular to the tangent space at each point
• Normalized to have unit length with respect to the ambient metric
• For orientable submanifolds, can be chosen consistently to define a global normal vector field
• Plays a crucial role in defining the second fundamental form and shape operator
• Used in the Gauss formula to decompose ambient covariant derivatives
• Essential in formulating the Gauss-Bonnet theorem for surfaces

Shape Operator and Weingarten Equation

Shape Operator Definition and Properties

• Shape operator quantifies how the unit normal vector changes along the submanifold
• Defined as $S_p(X) = -\nabla_X \nu$ where $\nabla$ is the ambient connection and $\nu$ is the unit normal
• Self-adjoint linear operator on the tangent space at each point
• Eigenvalues of the shape operator are the principal curvatures
• Trace of the shape operator gives the mean curvature
• Determinant of the shape operator relates to the Gaussian curvature for surfaces

Weingarten Equation and Its Implications

• Weingarten equation relates the differential of the Gauss map to the shape operator
• Expressed as $dG_p(X) = -S_p(X)$ for tangent vectors $X$
• Establishes the fundamental link between the extrinsic and intrinsic geometry
• Allows computation of the shape operator through the Gauss map
• Crucial in deriving the Gauss-Codazzi equations
• Generalizes to higher codimension using the second fundamental form

Second Fundamental Form and Curvature Analysis

• Second fundamental form measures the extrinsic curvature of a submanifold
• Defined as $II(X, Y) = \langle S(X), Y \rangle = -\langle \nabla_X Y, \nu \rangle$ for tangent vectors $X$ and $Y$
• Symmetric bilinear form on the tangent space at each point
• Local expression in coordinates: $II = h_{ij} dx^i dx^j$ where $h_{ij}$ are the coefficients
• Relates to the shape operator via $\langle S(X), Y \rangle = II(X, Y)$
• Used to classify points on a surface as elliptic, hyperbolic, or parabolic