Bump functionIn mathematics, a bump function (also called a test function) is a function on a Euclidean space which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain forms a vector space, denoted or The dual space of this space endowed with a suitable topology is the space of distributions. The function given by is an example of a bump function in one dimension.
TangentIn geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f(c), where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
Euclidean distanceIn mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.
4-manifoldIn mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).
Kähler manifoldIn mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.
Haken manifoldIn mathematics, a Haken manifold is a compact, P2-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface. A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken.
Pseudo-Euclidean spaceIn mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, ..., en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vector x. For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, q is an isotropic quadratic form, otherwise it is anisotropic.
Euclidean planeIn mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E2. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.
Hyperkähler manifoldIn differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979. Equivalently, a hyperkähler manifold is a Riemannian manifold of dimension whose holonomy group is contained in the compact symplectic group Sp(n).
3-manifoldIn mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. A topological space is a 3-manifold if it is a second-countable Hausdorff space and if every point in has a neighbourhood that is homeomorphic to Euclidean 3-space.
Glossary of Riemannian and metric geometryThis is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below. Connection Curvature Metric space Riemannian manifold See also: Glossary of general topology Glossary of differential geometry and topology List of differential geometry topics Unless stated otherwise, letters X, Y, Z below denote metric spaces, M, N denote Riemannian manifolds, |xy| or denotes the distance between points x and y in X.
Riemann sphereIn mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as well-behaved.
