A harmonic map is a mathematical concept that generalizes the notion of a harmonic function to maps between Riemannian manifolds (spaces equipped with a geometric structure).

Definition:

A harmonic map is a smooth function $f: (M, g_M) \to (N, g_N)$ between two Riemannian manifolds $M$ and $N$ that minimizes the energy functional:

$E(f) = \int_M |df|^2 \, d\text{vol}_M$

Here:

• $M$ and $N$ are Riemannian manifolds,
• $g_M$ and $g_N$ are their respective Riemannian metrics,
• $df$ is the differential of the map $f$, which is a section of the pullback bundle $f^*TN$, the tangent bundle of $N$ pulled back to $M$,
• $|df|^2$ is the pointwise norm of the differential of $f$, and
• $d\text{vol}_M$ represents the volume element on $M$.

Harmonic Map Equations:

The condition that $f$ is harmonic is equivalent to the fact that it satisfies the harmonic map equation (a second-order partial differential equation):

$\tau(f) = 0$

where $\tau(f)$ is the tension field of $f$, defined as:

$\tau(f) = \text{trace} \left( \nabla df \right)$

Here, $\nabla df$ is the covariant derivative of the differential of $f$, and $\tau(f)$ is a vector field on $M$ that measures how the map $f$ deviates from being "locally constant" or "harmonic."

In simpler terms, a map is harmonic if it "minimizes" the energy required to map the manifold $M$ into $N$, and the tension field describes the deviation from this minimum.

Intuition:

• For example, if $M$ and $N$ are both Euclidean spaces, a harmonic map can be seen as the solution to a boundary value problem where the map minimizes the total distortion (energy) between the two spaces.
• Harmonic maps are the "natural" generalization of harmonic functions, and they play an important role in many areas of geometry and physics, especially in the study of minimal surfaces and the theory of gauge fields.

Example:

Consider the simplest case: the map $f$ from a curve (1-dimensional manifold) in Euclidean space to another curve. If this map is harmonic, it will correspond to a geodesic in the target space, because geodesics minimize the energy.

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Let me know if you'd like a deeper dive into any of the above aspects or examples! Here are 5 related questions you could explore further:

1. How are harmonic maps used in the study of minimal surfaces?
2. What are the properties of the tension field $\tau(f)$ in harmonic maps?
3. Can you explain the energy functional in more detail?
4. What is the relationship between harmonic maps and harmonic functions?
5. How does harmonic map theory apply to physics, especially in string theory or gauge theory?

Tip: In harmonic map theory, energy minimization is a powerful concept—look for physical systems or mathematical problems where minimizing energy can lead to optimal configurations, like in physics or geometry.