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The inner product in Rⁿ (the familiar Euclidean dot product) allows you to define lengths of vectors and angles between vectors. For example, if a and b are vectors in Rⁿ, then a² is the length squared of the vector, and a*b gives the cosine of the angle between them (a*b=||a||||b|| cos θ). The inner product is a concept from linear algebraLinear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is wi which can be defined for any vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for. Since the tangent bundleIn mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. The tangent bundle of manifold M is usually denot of a smooth manifold (or indeed, any vector bundleIn mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topolo over a manifold) is, considered pointwise, just a vector space, it too can carry an inner product. If the inner product on the tangent space of a manifold is smoothly defined, then concepts that were defined only pointwise at each tangent space can be integrated, to yield analogous notions over finite regions of the manifold. In this context, the tangent space can be thought of as an infinitesimal translation on the manifold. Thus, the inner product on the tangent space gives the length of an infinitesimal translation. The integralThis article deals with the concept of an integral in mathematical calculus. For other meanings of "integral" see integration. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. Unlike the process of differe of this length gives the length of a curve on the manifold. To pass from a linear algebraic concept to a differential geometric one, the smoothness requirement is important, in many instances.

Every smooth submanifold of Rⁿ has an induced Riemannian metric: the inner product on each tangent space is the restriction of the inner product on Rⁿ. In fact, as it follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way. In particular one could define Riemannian manifold as a metric space which is isometric to a smooth submanifold of Rⁿ with the induced intrinsic metric. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry.

Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of positive-definite quadratic forms on the tangent bundle. Then one has to work to show that it can be turned to a metric space:

If γ : [a, b] → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L(γ) by

(Note that γ'(t) is an element of the tangent space to M at the point γ(t); ||.|| denotes the norm resulting from the given inner product on that tangent space.)

With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as

d(x,y) = inf{ L(γ) : γ is a continuously differentiable curve joining x and y}.

Even though Riemannian manifolds are usually "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points along shortest paths.