# What kind of isometry? A metric tensor "respects" the foliation?

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In summary, for a foliation of leaves with codimension one on a Riemannian manifold ##M## with metric ##g##, a flat coordinate chart can be found such that setting ##y## to a constant uniquely labels each leaf. The dimension of ##M## is ##n## and the metric can be expressed in terms of a coordinate basis as ##g=g_{ab}dx^a \otimes dx^b##. If ##\frac{\partial}{\partial y} g_{ab} = 0## in any flat chart, then the foliation is respected by the metric. This type of isometry is called a Killing vector and implies that the manifold is diffeomorphic to ##Q \times \
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Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf (hypersurface) uniquely.

Let the dimension of the manifold ##M## be ##n## and let latin indicies run from ##1## to ##n## and greek run from ##1## to ##n-1##.

In the chart we can introduce a coordinate basis and express the metric as ##g=g_{ab}dx^a \otimes dx^b##. Now suppose that
$$\frac{\partial}{\partial y} g_{ab} = 0$$
in _any_ flat chart for the foliation.

Since the ##y##-curves defined by ##x^\mu## to constant are only defined within each flat chart for the foliation they are not globally defined (they might be disconnected and be at completely different angles from their respective leaves). And hence ##frac{\partial}{\partial y}## is not a globally defined killing vector.

But what are this kind of isometry then kalled? Somehow the metric ##g## respects the foliation. Are there any equivalent definitions of this situation?

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

What you're doing is a bit weird, in the sense that ##y## is not a coordinate, but a label for the leaves, which can be considered a global parameter. The leaves ##\Lambda_y## need not admit a global chart. A better notation for the chart would be
$$(U_y \subseteq \Lambda_y, (x^i_y))$$
Corresponding to the foliation you have a distribution locally spanned by the pushforward of the coordinate frames to ##M##. If ##\frac{d}{d y}## commutes with this distribution, then locally you can find an (adapted) chart on ##M## s.t. the y becomes a coordinate and the other coordinates coincide with the ones on the leaves. This distinction should be made.
In the former case, the equation you actually wrote down is
$$\frac{d}{d y} g \downharpoonright_{\Lambda_y} = 0 \, ,$$
which is trivial. In the latter case you wrote
$$\mathcal{L}_{\frac{d}{d y}} g = 0 \, ,$$
which is exactly the condition for it to be a Killing vector field and might not be true in general. If it is, the metric is preserved by the flow of the vector field ##\frac{d}{d y}## and, again assuming the foliation exists, it is enough to study one of the leaves and then the manifold ##M## is the orbit of the action of the flow on the leaf. Note that this implies that your manifold is diffeomorphic to ##Q \times \mathbb R## or
##Q \times S^1##.

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## 1. What is an isometry?

An isometry is a transformation that preserves distance and angles between objects. In other words, if two objects are congruent before the transformation, they will remain congruent after the transformation.

## 2. What is a metric tensor?

A metric tensor is a mathematical object that defines the distance between two points in a space. It is used to measure the length of a curve, the angle between two curves, and other geometric properties.

## 3. What does it mean for a metric tensor to "respect" a foliation?

In this context, "respecting" a foliation means that the metric tensor is compatible with the structure of the foliation. This means that the metric tensor remains unchanged when moving along the direction of the foliation, and that the metric tensor can be decomposed into a sum of tensors that act independently on each leaf of the foliation.

## 4. How can you tell if a metric tensor respects a foliation?

A metric tensor respects a foliation if it satisfies the compatibility conditions known as the "Gauss-Codazzi equations." These equations relate the curvature of the space to the intrinsic and extrinsic geometry of the foliation.

## 5. Why is it important for a metric tensor to respect a foliation?

If a metric tensor respects a foliation, it allows for the use of powerful mathematical tools, such as the Gauss-Codazzi equations, to study the geometry of the space. This can provide insights into the structure and properties of the space, as well as aid in the development of theories and models in physics and other scientific fields.

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