Structure defining distance on a manifold

Metric tensor

Metric tensor

In the <a href="/en/Mathematics" title="Mathematics" class="wl">mathematical</a> field of <a href="/en/Differential_geometry" title="Differential geometry" class="wl">differential geometry</a>, a metric tensor (or simply metric) is an additional <a href="/en/Mathematical_structure" title="Mathematical structure" class="wl">structure</a> on a <a href="/en/Manifold" title="Manifold" class="wl">manifold</a> M (such as a <a href="/en/Surface_(mathematics)" title="Surface (mathematics)" class="wl">surface</a>) that allows defining distances and angles, just as the <a href="/en/Inner_product" title="Inner product" class="mw-redirect wl">inner product</a> on a <a href="/en/Euclidean_space" title="Euclidean space" class="wl">Euclidean space</a> allows defining distances and angles there. More precisely, a metric tensor at a point p of M is a <a href="/en/Bilinear_form" title="Bilinear form" class="wl">bilinear form</a> defined on the <a href="/en/Tangent_space" title="Tangent space" class="wl">tangent space</a> at p (that is, a <a href="/en/Bilinear_function" title="Bilinear function" class="mw-redirect wl">bilinear function</a> that maps pairs of <a href="/en/Tangent_vector" title="Tangent vector" class="wl">tangent vectors</a> to <a href="/en/Real_number" title="Real number" class="wl">real numbers</a>), and a metric tensor on M consists of a metric tensor at each point p of M that varies smoothly with p.

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A metric tensor g is positive-definite if g(v, v) &gt; 0 for every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a <a href="/en/Riemannian_manifold" title="Riemannian manifold" class="wl">Riemannian manifold</a>. Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold M, the length of a smooth curve between two points p and q can be defined by integration, and the <a href="/en/Metric_(mathematics)" title="Metric (mathematics)" class="mw-redirect wl">distance</a> between p and q can be defined as the <a href="/en/Infimum" title="Infimum" class="mw-redirect wl">infimum</a> of the lengths of all such curves; this makes M a <a href="/en/Metric_space" title="Metric space" class="wl">metric space</a>. Conversely, the metric tensor itself is the <a href="/en/Derivative" title="Derivative" class="wl">derivative</a> of the distance function (taken in a suitable manner).[<a href="/en/Wikipedia:Citation_needed" title="Wikipedia:Citation needed" class="wl">citation needed</a>]

While the notion of a metric tensor was known in some sense to mathematicians such as <a href="/en/Gauss" title="Gauss" class="mw-redirect wl">Gauss</a> from the early 19th century, it was not until the early 20th century that its properties as a <a href="/en/Tensor" title="Tensor" class="wl">tensor</a> were understood by, in particular, <a href="/en/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro" class="wl">Gregorio Ricci-Curbastro</a> and <a href="/en/Tullio_Levi-Civita" title="Tullio Levi-Civita" class="wl">Tullio Levi-Civita</a>, who first codified the notion of a tensor. The metric tensor is an example of a <a href="/en/Tensor_field" title="Tensor field" class="wl">tensor field</a>.

The components of a metric tensor in a <a href="/en/Coordinate_basis" title="Coordinate basis" class="mw-redirect wl">coordinate basis</a> take on the form of a <a href="/en/Symmetric_matrix" title="Symmetric matrix" class="wl">symmetric matrix</a> whose entries transform <a href="/en/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors" class="wl">covariantly</a> under changes to the coordinate system. Thus a metric tensor is a covariant <a href="/en/Symmetric_tensor" title="Symmetric tensor" class="wl">symmetric tensor</a>. From the <a href="/en/Coordinate-free" title="Coordinate-free" class="wl">coordinate-independent</a> point of view, a metric tensor field is defined to be a <a href="/en/Nondegenerate_form" title="Nondegenerate form" class="mw-redirect wl">nondegenerate</a> <a href="/en/Symmetric_bilinear_form" title="Symmetric bilinear form" class="wl">symmetric bilinear form</a> on each tangent space that varies <a href="/en/Smooth_function" title="Smooth function" class="mw-redirect wl">smoothly</a> from point to point.

In the mathematical field of differential geometry, a metric tensor is an additional structure on a manifold M that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there.  More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p , and a metric tensor on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g &gt; 0 for every nonzero vector v.  A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold.  Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold.  On a Riemannian manifold M, the length of a smooth curve between two points p and q can be defined by integration, and the distance between p and q can be defined as the infimum of the lengths of all such curves; this makes M a metric space.  Conversely, the metric tensor itself is the derivative of the distance function . While the notion of a metric tensor was known in some sense to mathematicians such as Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor.  The metric tensor is an example of a tensor field. 