1.4 Differential forms on manifolds

Differential forms are powerful tools in manifold theory, generalizing functions to accept vector inputs and produce scalar outputs. They're crucial for understanding integration, cohomology, and key theorems in differential geometry.

This section explores the definition and properties of differential forms, operations like exterior derivative and wedge product, and their applications in integration and cohomology. It also covers important results like Stokes' theorem and concepts of orientation and volume forms.

Differential Forms and Operations

Definition and Properties of Differential Forms

Top images from around the web for Definition and Properties of Differential Forms

• 

  Exterior algebra - Wikipedia View original

  Is this image relevant?

• 

  differential geometry - How to visualize $1$-forms and $p$-forms? - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Operations and Actions of Lie Groups on Manifolds View original

  Is this image relevant?

• 

  Exterior algebra - Wikipedia View original

  Is this image relevant?

• 

  differential geometry - How to visualize $1$-forms and $p$-forms? - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Properties of Differential Forms

• 

  Exterior algebra - Wikipedia View original

  Is this image relevant?

• 

  differential geometry - How to visualize $1$-forms and $p$-forms? - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Operations and Actions of Lie Groups on Manifolds View original

  Is this image relevant?

• 

  Exterior algebra - Wikipedia View original

  Is this image relevant?

• 

  differential geometry - How to visualize $1$-forms and $p$-forms? - Mathematics Stack Exchange View original

  Is this image relevant?

1 of 3

• Differential form is a smooth section of the exterior algebra of the cotangent bundle of a manifold
• Generalizes the concept of a function to accept vector inputs and produce scalar outputs
• Differential forms are antisymmetric multilinear maps that take tangent vectors as input and produce real numbers as output
• The set of all differential forms on a manifold $M$ is denoted by $\Omega^k(M)$, where $k$ is the degree of the form
• Differential forms of degree 0 are smooth functions on the manifold

Exterior Derivative and Wedge Product

• Exterior derivative is an operator that maps $k$-forms to $(k+1)$-forms, denoted by $d: \Omega^k(M) \to \Omega^{k+1}(M)$
• For a smooth function $f$, the exterior derivative $df$ is the differential of $f$, which is a 1-form
• The exterior derivative satisfies the property $d^2 = 0$, meaning that applying the exterior derivative twice always yields zero
• Wedge product is an operation that combines two differential forms to create a new differential form of higher degree
• For a $k$-form $\alpha$ and an $l$-form $\beta$, their wedge product is a $(k+l)$-form denoted by $\alpha \wedge \beta$
• The wedge product is associative and anticommutative, i.e., $\alpha \wedge \beta = (-1)^{kl} \beta \wedge \alpha$

Integration on Manifolds

• Integration on manifolds generalizes the concept of integration from Euclidean spaces to manifolds
• To integrate a differential form over a manifold, we need to have a notion of orientation and a volume form
• Integration of a $k$-form $\omega$ over a $k$-dimensional oriented submanifold $S$ is denoted by $\int_S \omega$
• The fundamental theorem of calculus relates the integral of a differential form over the boundary of a manifold to the integral of its exterior derivative over the manifold itself

Cohomology and Theorems

De Rham Cohomology

• De Rham cohomology is a cohomology theory based on differential forms and the exterior derivative
• The $k$-th de Rham cohomology group of a manifold $M$ is defined as the quotient space $H^k(M) = \ker d_k / \operatorname{im} d_{k-1}$
• Elements of the kernel of the exterior derivative are called closed forms, while elements of the image are called exact forms
• The dimension of the $k$-th de Rham cohomology group is a topological invariant of the manifold, known as the $k$-th Betti number
• De Rham cohomology provides a way to study the global properties of a manifold using differential forms

Stokes' Theorem

• Stokes' theorem is a fundamental result that relates the integral of a differential form over the boundary of a manifold to the integral of its exterior derivative over the manifold itself
• For an oriented manifold $M$ with boundary $\partial M$ and a differential form $\omega$, Stokes' theorem states that $\int_M d\omega = \int_{\partial M} \omega$
• Stokes' theorem generalizes various integral theorems, such as the fundamental theorem of calculus, Green's theorem, and the divergence theorem
• Stokes' theorem has important applications in physics, such as in electromagnetism and fluid dynamics, where it relates the flow of a vector field through a surface to the circulation around its boundary

Orientation and Volume

Orientation of Manifolds

• Orientation is a global property of a manifold that allows for a consistent choice of "clockwise" or "counterclockwise" direction
• A manifold is orientable if it admits a consistent choice of orientation across all of its charts
• An oriented manifold is a manifold with a chosen orientation
• Orientation is crucial for defining integration on manifolds and for stating results like Stokes' theorem
• Examples of orientable manifolds include the circle, the sphere, and the torus, while examples of non-orientable manifolds include the Möbius strip and the Klein bottle

Volume Form and Integration

• A volume form on an oriented manifold is a nowhere-vanishing differential form of top degree that is compatible with the orientation
• In local coordinates $(x^1, \ldots, x^n)$, a volume form can be written as $\omega = f(x) dx^1 \wedge \cdots \wedge dx^n$, where $f(x)$ is a positive smooth function
• The volume form allows for the definition of integration on the manifold, where the integral of a function $g$ over the manifold $M$ is given by $\int_M g \omega$
• The volume of a compact oriented manifold $M$ is defined as the integral of the constant function 1 with respect to the volume form, i.e., $\operatorname{Vol}(M) = \int_M \omega$
• The volume form and integration on manifolds play a crucial role in various areas of mathematics and physics, such as in differential geometry, topology, and general relativity