Cotangent Space

The Cotangent Space at a point $p$ in a differentiable manifold $M$ is the dual space to the Tangent Space $T_pM$. It is denoted by $T_p^M$ or $T_p^(M)$.

The Cotangent Space $T_p^*M$ is composed of all linear maps that map from $T_pM$ to the real numbers. As such, elements of the cotangent space are often viewed as differential 1-forms.

The collection of all cotangent spaces at every point in the manifold forms the Cotangent Bundle $T^*M$, a concept central to differential forms and symplectic geometry.

See also

• Cotangent bundle
• Transversality
• Tangent space, (bundle)
• Morse theory
• Floer homology

(edit) Topics in Topological Data Analysis (📁) Background Point-set topology • Algebraic topology Manifolds Manifold • Configuration space • Differentiability • Vector space (bundle) • Tangent space (bundle) • Cotangent space (bundle) • Vector field (Flow, Critical points) • Braid • Transversality • Whitney embedding theorem • Stratified space Complexes CW Complex • Δ-Complex • Regular Complex (Simplicial, Cubical, Vietoris-Rips, Čech, Witness, Flag, Strategy • Decision Task) • Nerve • Phylogenetic tree • Discretized graph configuration space • State complex Euler Characteristic Euler characteristic • Curvature • Nonvanishing vector field • Fixed point index • Tame topology • Euler calculus • Target enumeration • Fubini theorem • Euler integral transform • Intrinsic volume • Gaussian random field Exact Sequences Chain homotopy • Snake lemma • Good pair • Mayer-Vietoris sequence • 5-lemma • Cellular homology • Coverage • Degree • Sperner's Lemma • Borsuk-Ulam theorem • Short exact sequence • Lefschetz index • Nash equilibrium • Hex • Barcode • Space of natural images • Zigzag persistence Homology Simplicial homology • Künneth theorem • Coefficient • Singular homology • Reduced homology • Čech homology • Relative homology • Excision Theorem • Local homology • Relation homology • Functoriality • Inverse kinematics • Winding number • Brouwer fixed point theorem Cohomology Duality • Cochain complex • Cohomology • Universal coefficient theorem • Poincaré duality • Alexander duality • Helly's Theorem • Numerical Euler integration • Forms • De Rham cohomology • Cup product • Helicity Theorem • Laplacian • Hodge Theorem • Cohomological coordination Morse Theory Morse homology • Excursion set • Definable functional • Euler-Fourier transform • Stratified Morse theory • Conley index • Forcing theory • Attractor-repeller pairs • Floer homology • Symplectic form • Hamiltonian field • Arnold's Conjecture • Lefschetz-Hopf theorem • Stratified Lefschetz-Morse Index • Discrete Morse theory • LS category • Unimodal decomposition Homotopy Fundamental group • Van-Kampen theorem • Covering space • Knot • Braid • Seifert conjecture • Higher homotopy groups • Biaxial nematics • Hurewicz Theorem • Whitehead Theorem • Eilenberg-MacLane space • Topological social choice • Fiber bundle • Section • Path space • Farber lemma • Sectional category • Topological complexity • Fibration • Homotopy extension property • Model category • Homotopy type theory Sheaves Cellular sheaf • Local cohomology • Fiber homology • Sheaf cohomology • Global sections theorem • Linear sheaf • Flow sheaf • Network coding sheaf • Information flow theorem • Čech cohomology • Preseaf • Stalk • Gluing axiom • PDE sheaf • Operations on sheaves • Vietoris mapping theorem • Sheaf-theoretic Nyquist-Shannon • Fourier-Sato transform • Cosheaf • Bézier curve • Barcodes • Interval Indecomposables • Étale Space • Leray spectral sequence • Spectral sheaf theory • Sheaf neural network Categorification Category • Groupoid • Monoid • Morphism • Equalizer • Functor • Kan complex • Nerve • Clustering • Natural transformation • Cone • Interleaving Limit • Terminal object • Product • Pullback • Homology Colimit • Initial object • Coproduct • Pushout • Cohomology Topological sheaf • Sheaf operations • Derived functors